quadratic inequalities homework

Solving Quadratic Inequalities

... and more ...

A Quadratic Equation (in Standard Form) looks like:

The above is an equation (=) but sometimes we need to solve inequalities like these:

Solving inequalities is very like solving equations ... we do most of the same things.

So this is what we do:

• find the "=0" points
• greater than zero (>0), or
• less than zero (<0)
• then pick a test value to find out which it is (>0 or <0)

Here is an example:

Example: x 2 − x − 6 < 0

x 2 − x − 6 has these simple factors (what luck!):

(x+2)(x−3) < 0

Firstly , let us find where it is equal to zero:

(x+2)(x−3) = 0

It is equal to zero when x = −2 or x = +3 because when x = −2, then (x+2) is zero or when x = +3, then (x−3) is zero

So between −2 and +3, the function will either be

• always greater than zero, or
• always less than zero

We don't know which ... yet!

Let's pick a value in-between (say x=0) and test it:

So between x=−2 and x=+3, the function is less than zero.

And that is the region we want, so...

x 2 − x − 6 < 0 in the interval (−2, 3)

Note: x 2 − x − 6 > 0   in the interval (−∞,−2) and (3, +∞)

Also try the Inequality Grapher .

What If It Doesn't Go Through Zero?

A "real world" example, a stuntman will jump off a 20 m building. a high-speed camera is ready to film him between 15 m and 10 m above the ground..

When should the camera film him?

We can use this formula for distance and time:

d = 20 − 5t 2

• d = distance above ground (m), and
• t = time from jump (seconds)

(Note: if you are curious about the formula, it is simplified from d = d 0 + v 0 t + ½a 0 t 2 , where d 0 =20 , v 0 =0 , and a 0 =−9.81 the acceleration due to gravity.)

OK, let's go.

First , let us sketch the question:

The distance we want is from 10 m to 15 m :

10 < d < 15

And we know the formula for d :

10 < 20 − 5t 2 < 15

Now let's solve it!

First, let's subtract 20 from both sides:

−10 < −5t 2 <−5

 Now multiply both sides by −(1/5). But because we are multiplying by a negative number, the inequalities will change direction ... read Solving Inequalities to see why.

2 > t 2 > 1

To be neat, the smaller number should be on the left, and the larger on the right. So let's swap them over (and make sure the inequalities still point correctly):

1 < t 2 < 2

 Lastly, we can safely take square roots, since all values are greater then zero:

√1 < t < √2

We can tell the film crew:

"Film from 1.0 to 1.4 seconds after jumping"

Higher Than Quadratic

The same ideas can help us solve more complicated inequalities:

Example: x 3 + 4 ≥ 3x 2 + x

First, let's put it in standard form:

x 3 − 3x 2 − x + 4 ≥ 0

This is a cubic equation (the highest exponent is a cube, i.e. x 3 ), and is hard to solve, so let us graph it instead:

The zero points are approximately :

And from the graph we can see the intervals where it is greater than (or equal to) zero:

• From −1.1 to 1.3, and
• From 2.9 on

In interval notation we can write:

Approximately: [−1.1, 1.3] U [2.9, +∞)

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Quadratic Inequalities Worksheets

Supercharge your high school students' solving skills with our printable quadratic inequalities worksheets. Represent the inequality as an equation, moving the terms to one side and equating it to zero, factor the equation and find the zeros to obtain break points or critical points, graph them on a number line, and determine the interval. Solving quadratic inequalities algebraically, graphically, and completing the table of signs, graphing the parabola and shading the solution region based on the inequality are the exercises presented in these pdfs. You now have the chance to test your skills with our free quadratic inequalities worksheets.

Printing Help - Please do not print quadratic inequalities worksheets directly from the browser. Kindly download them and print.

Solving Quadratic Inequalities Algebraically | Level 1

With either -1 or 1 as leading coefficients and integer break points, these printable quadratic inequalities worksheets get high school students transforming quadratic inequalities into quadratic equations and factoring them to solve the inequality.

• Download the set

Solving Quadratic Inequalities Algebraically | Level 2

Triumph in solving quadratic inequalities whose leading coefficients are integers by finding the factors, equating them to zero, determining their roots, and finding the solution range in this compilation of quadratic inequalities worksheet pdfs.

Solving Quadratic Inequalities Algebraically | Level 3

Instruct high school students to express the quadratic inequalities in standard form by moving the expressions on one side and equating to 0, factor them, determine their roots, to find the critical point and figure out the solution interval.

Solving Quadratic Inequalities | Table Method

Factor the quadratic inequalities and list out the intervals in the table of signs. Take a test value from each interval and apply on both the factors. Complete the table with the resultant plus or minus sign. Select the solution range that satisfies the inequality.

Graphing and Solving Inequalities

Check if the quadratic inequality is inclusive or strict. Graph the parabola y = f(x) for the quadratic inequality f(x) ≤ 0 or f(x) ≥ 0. Find the vertex and identify the values of x for which the part of the parabola will either be negative or positive depending on the inequalities.

Graphing Quadratic Inequalities

High school students plot x-intercepts, figure out the axis of symmetry and the vertex of the parabola, determine the direction, and illustrate the inequality using dotted or solid lines. Shade the parabola below or above the x-axis, inside or outside the parabola based on the solution.

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» One Step Inequalities

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» Multi Step Inequalities

» Graphing Linear Equations

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Graphing Quadratic Inequalities Calculator

What are quadratic inequalities, how do i solve quadratic inequalities by graphing, how to use this graphing quadratic inequalities calculator, examples of graphing quadratic inequalities.

Thanks to Omni's graphing quadratic inequalities calculator, you will beat your homework assignment in no time! We will teach you how to solve quadratic inequalities by graphing parabolas and how it can help you avoid using the dreaded quadratic formula . Let's go!

🙋 However, in case you want to have a look, we do have a dedicated quadratic formula calculator !

A quadratic inequality is an expression that involves a quadratic trinomial (so a polynomial of degree 2 ), which we compare to some other expression, most often to number zero. For instance:

ax² + bx + c > 0

Of course, other inequality signs ≥ , ≤ , or < are also possible.

🙋 To learn more about various aspects of quadratic equations, make sure to visit our factoring trinomials calculator .

To solve a quadratic inequality ax² + bx + c > d :

• Draw the line y = d .
• Determine the points where the parabola ax² + bx + c crosses/touches this line. To find them, solve ax² + bx + (c - d) = 0 . If there's no solution, your parabola is entirely above or below the line.
• The arms go upwards if a > 0 .
• The arms go downwards if a < 0 .
• Because our inequality features the > sign, we check where the parabola is strictly above the line .

Nothing can be easier than using Omni's graphing quadratic inequalities calculator! Here's a brief instruction:

• Input the coefficients of your quadratic inequality into their fields.
• Make sure to adjust the inequality sign to your needs.
• The results appear immediately at the bottom of our graphing quadratic inequalities calculator!
• Our tool gives the results in terms of a graph and also as an interval .
• You may use our inequality to interval notation calculator to rewrite the solution as inequalities .

You see? As promised, with our tool bookmarked, you'll never again struggle with the question of how to graph quadratic inequalities!

In this final section, we include some examples of graphing solutions to quadratic inequalities so that you can verify if you already understand perfectly well how to graph quadratic inequalities.

Let's discuss how to graph the quadratic inequality -x² + 3x - 2 ≥ 0 .

First, we draw the line y = 0 . Obviously, it coincides with the horizontal axis. Let's move on to the parabola. Since a = -1 , we see the parabola will have its arms pointing downwards. We need to solve

-x² + 3x - 2 = 0 ,

to determine where it crosses the horizontal axis. Using one of the many methods of solving quadratic equations, we find that x = 1 or x = 2 .

Since we have the ≥ sign, we're interested in points where the parabola is above or precisely at the axis. It is now easy to conclude that our inequality is satisfied when 1 ≤ x ≤ 2 , or, in interval notation, x ∊ [1, 2] .

Let's graph the quadratic inequality:

x² + 2x + 3 > 2

As before, we draw the horizontal line corresponding to the right-hand side of our inequality: y = 2 . We see that we have a > 0 in our quadratic trinomial, so the arms will go upward. To determine precisely where the parabola crosses the line, we need to solve the equation:

x² + 2x + 3 = 2 .

Adding -2 to both sides, we get:

x² + 2x + 1 = 0

We recognize a perfect square trinomial! We can further transform it via the short multiplication formula to:

(x + 1)² = 0

And, finally, we get:

Therefore, the parabola does not cross the line but touches it (and gently!) at x = -1 . Since we deal with the inequality

x² + 2x + 3 > 2 ,

we're looking for arguments where the parabola is strictly above the line. Hence, our inequality is satisfied for each argument x apart from the one where the parabola touches the line, so for x ∊ ℝ \ {-1} .

Finally, we discuss how to solve 2x² + 3x + 4 < 1 by graphing.

First, of course, we draw the horizontal line y = 1 . Next, we solve

2x² + 3x + 4 = 1 .

So, adding -1 to both sides, we get:

2x² + 3x + 3 = 0 .

We start applying the quadratic formula:

Δ = 3² - 4 × 3 × 3 = 9 - 12 = -3 .

Since the discriminant Δ is negative, we know the equation in question has no (real) solutions, and so our parabola does not cross/touch the horizontal line. Since this parabola points upwards ( a = 2 ), it follows that it lies entirely above the line. And our inequality asks for points where the parabola is strictly below the line ('<' sign). We conclude that there are no solutions: x ∊ ∅ .

You can use the graphing quadratic inequalities calculator to go through all of these examples, as well as generate your own. Remember that practice makes perfect!

How do I graph solutions to quadratic inequalities?

You can graph solutions to quadratic inequalities as intervals or unions of (at most two) intervals on the number line. This method is very similar to graphing absolute value inequalities. However, to precisely find the boundary points of the intervals, you will have to solve a quadratic equation, using, e.g., the quadratic formula.

How do I graph the system of quadratic inequalities?

To graph the system of quadratic inequalities, you need to carefully graph these inequalities one-by-one on a common plot, and then find all the arguments where all the inequalities are satisfied.

How do I solve x² < 1 by graphing?

• Draw the parabola y = x² : it touches the horizontal axis at x = 0 , and its arms go upwards.
• Draw the horizontal line y = 1 .
• Find where the parabola is strictly below the line: for -1 < x < 1 .
• That's it! Your inequality holds for x ∊ (-1, 1) , and you solved it by graphing!

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MATH 1111 - College Algebra: 3.2 Quadratic Functions and Quadratic Inequalities

• 1.1 Sets and Set Operations
• 1.2 Linear Equations and Inequalities
• 1.3 Systems of Linear Equations
• 1.4 Polynomials; Operations with Polynomials
• 1.5 Factoring Polynomials
• 1.6 Quadratic Equations
• 1.7 Rational Expressions and Equations
• 1.8 Complex Numbers
• 2.1 Cartesian Coordinates/Relations
• 2.2 Intro to Functions
• 2.3 Operations with Functions
• 2.4 Graph of Functions
• 3.1 Linear Functions

3.2 Quadratic Functions and Quadratic Inequalities

• 4.1 Finding Zeros of Polynomial Functions
• 4.2 Graphing Polynomial Functions
• 4.3 Rational Functions
• 4.4 Rational Inequalities
• 5.1 Composition of Functions
• 5.2 Inverse Functions
• 5.3 Introduction to Exponential and Logarithmic Functions

At the end of this section students will be able to:

• Determine the general and standard form of a quadratic function
• Graph a quadratic function using its vertex and x-and y-intercepts
• Graph a quadratic function using transformation techniques
• Solve applications using quadratic functions
• Solve quadratic inequalities

Required Reading

2.3 Quadratic Functions

Stitz-Zeager College Algebra  - pages  188-199

2.4 Inequalities with Absolute Value and Quadratic Functions    

Stitz-Zeager College Algebra  - pages  208-217

Practice Exercises

Quadratic Functions

Stitz-Zeager College Algebra  - pages 200-201

Answers to practice exercises can be found on pages 203-207.

Quadratic Inequalities

Stitz-Zeager College Algebra  - pages 220

Answers to practice exercises can be found on page 222.

Supplemental Resources

Graphs of Quadratic Functions  (tutorial):  West Texas A&M University Virtual Math Lab (College Algebra Tutorial 34)

Quadratic Inequalities (tutorial):  West Texas A&M University Virtual Math Lab (College Algebra Tutorial 23A)

Find the Standard Form (Vertex Form) and the Vertex of a Quadratic Function:

Graphing a Parabola Using its Vertex and x-intercepts:

Graphing Quadratic Functions Using Transformations Techniques:

Solving Quadratic Inequalities: 

Solving Applications of Quadratic Functions: 

• << Previous: 3.1 Linear Functions
• Next: Module 4 >>
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Unit 14: Quadratic functions & equations

About this unit.

We've seen linear and exponential functions, and now we're ready for quadratic functions. We'll explore how these functions and the parabolas they produce can be used to solve real-world problems.

Intro to parabolas

• Parabolas intro (Opens a modal)
• Interpreting a parabola in context (Opens a modal)
• Interpret a quadratic graph (Opens a modal)
• Parabolas intro Get 3 of 4 questions to level up!
• Interpret parabolas in context Get 3 of 4 questions to level up!
• Interpret a quadratic graph Get 3 of 4 questions to level up!

Solving and graphing with factored form

• Zero product property (Opens a modal)
• Graphing quadratics in factored form (Opens a modal)
• Quadratic word problems (factored form) (Opens a modal)
• Zero product property Get 3 of 4 questions to level up!
• Graph quadratics in factored form Get 3 of 4 questions to level up!
• Quadratic word problems (factored form) Get 3 of 4 questions to level up!

Solving by taking the square root

• Solving quadratics by taking square roots (Opens a modal)
• Solving quadratics by taking square roots examples (Opens a modal)
• Quadratics by taking square roots: strategy (Opens a modal)
• Solving quadratics by taking square roots: with steps (Opens a modal)
• Solving simple quadratics review (Opens a modal)
• Quadratics by taking square roots (intro) Get 3 of 4 questions to level up!
• Quadratics by taking square roots Get 3 of 4 questions to level up!
• Quadratics by taking square roots: strategy Get 3 of 4 questions to level up!
• Quadratics by taking square roots: with steps Get 3 of 4 questions to level up!

Vertex form

• Vertex form introduction (Opens a modal)
• Graphing quadratics: vertex form (Opens a modal)
• Quadratic word problems (vertex form) (Opens a modal)
• Graph quadratics in vertex form Get 3 of 4 questions to level up!
• Quadratic word problems (vertex form) Get 3 of 4 questions to level up!

Solving quadratics by factoring

• Solving quadratics by factoring (Opens a modal)
• Solving quadratics by factoring: leading coefficient ≠ 1 (Opens a modal)
• Solving quadratics using structure (Opens a modal)
• Quadratic equations word problem: triangle dimensions (Opens a modal)
• Quadratic equations word problem: box dimensions (Opens a modal)
• Solving quadratics by factoring review (Opens a modal)
• Quadratics by factoring (intro) Get 3 of 4 questions to level up!
• Quadratics by factoring Get 3 of 4 questions to level up!
• Solve equations using structure Get 3 of 4 questions to level up!

The quadratic formula

• The quadratic formula (Opens a modal)
• Understanding the quadratic formula (Opens a modal)
• Worked example: quadratic formula (example 2) (Opens a modal)
• Worked example: quadratic formula (negative coefficients) (Opens a modal)
• Using the quadratic formula: number of solutions (Opens a modal)
• Quadratic formula review (Opens a modal)
• Discriminant review (Opens a modal)
• Quadratic formula Get 3 of 4 questions to level up!
• Number of solutions of quadratic equations Get 3 of 4 questions to level up!

Completing the square intro

• Completing the square (Opens a modal)
• Worked example: Completing the square (intro) (Opens a modal)
• Worked example: Rewriting expressions by completing the square (Opens a modal)
• Worked example: Rewriting & solving equations by completing the square (Opens a modal)
• Completing the square (intro) Get 3 of 4 questions to level up!
• Completing the square (intermediate) Get 3 of 4 questions to level up!

More on completing the square

• Solve by completing the square: Integer solutions (Opens a modal)
• Solve by completing the square: Non-integer solutions (Opens a modal)
• Worked example: completing the square (leading coefficient ≠ 1) (Opens a modal)
• Solving quadratics by completing the square: no solution (Opens a modal)
• Proof of the quadratic formula (Opens a modal)
• Solving quadratics by completing the square (Opens a modal)
• Completing the square review (Opens a modal)
• Quadratic formula proof review (Opens a modal)
• Solve equations by completing the square Get 3 of 4 questions to level up!
• Completing the square Get 3 of 4 questions to level up!

Strategizing to solve quadratic equations

• Strategy in solving quadratic equations (Opens a modal)
• Strategy in solving quadratics Get 3 of 4 questions to level up!

Quadratic standard form

• Finding the vertex of a parabola in standard form (Opens a modal)
• Graphing quadratics: standard form (Opens a modal)
• Quadratic word problem: ball (Opens a modal)
• Graph quadratics in standard form Get 3 of 4 questions to level up!
• Quadratic word problems (standard form) Get 3 of 4 questions to level up!

Features & forms of quadratic functions

• Forms & features of quadratic functions (Opens a modal)
• Worked examples: Forms & features of quadratic functions (Opens a modal)
• Vertex & axis of symmetry of a parabola (Opens a modal)
• Finding features of quadratic functions (Opens a modal)
• Interpret quadratic models: Factored form (Opens a modal)
• Interpret quadratic models: Vertex form (Opens a modal)
• Graphing quadratics review (Opens a modal)
• Creativity break: How does creativity play a role in your everyday life? (Opens a modal)
• Features of quadratic functions: strategy Get 3 of 4 questions to level up!
• Features of quadratic functions Get 3 of 4 questions to level up!
• Graph parabolas in all forms Get 3 of 4 questions to level up!
• Interpret quadratic models Get 3 of 4 questions to level up!

Comparing quadratic functions

• Comparing features of quadratic functions (Opens a modal)
• Comparing maximum points of quadratic functions (Opens a modal)
• Compare quadratic functions Get 3 of 4 questions to level up!

Transforming quadratic functions

• Intro to parabola transformations (Opens a modal)
• Shifting parabolas (Opens a modal)
• Scaling & reflecting parabolas (Opens a modal)
• Quadratic functions & equations: FAQ (Opens a modal)
• Shift parabolas Get 3 of 4 questions to level up!
• Scale & reflect parabolas Get 3 of 4 questions to level up!

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11: Quadratic Equations and Applications

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Learning Objectives

By the end of this chapter, the student should be able to

• Solve quadratics by the square root property, completing the square, and using the quadratic formula
• Graph a quadratic function by using properties or transformations
• Solve quadratic inequalities by graphing, or algebraically
• Find the extreme value of a quadratic function
• Solve applications and functions using quadratic functions

We might recognize a quadratic equation from the factoring chapter as a trinomial equation . Although, it may seem that they are the same, they aren’t the same. Trinomial equations are equations with any three terms. These terms can be any three terms where the degree of each term can vary. On the other hand, quadratic equations are equations with specific degrees on each term.

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• x^2+2x+1=3x-10
• 2x^2+4x-6=0
• How do you calculate a quadratic equation?
• To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
• What is the quadratic formula?
• The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b ± √(b^2 - 4ac)) / (2a)
• Does any quadratic equation have two solutions?
• There can be 0, 1 or 2 solutions to a quadratic equation. If the discriminant is positive there are two solutions, if negative there is no solution, if equlas 0 there is 1 solution.
• What is quadratic equation in math?
• In math, a quadratic equation is a second-order polynomial equation in a single variable. It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a ≠ 0.
• How do you know if a quadratic equation has two solutions?
• A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive.

quadratic-equation-calculator

• High School Math Solutions – Quadratic Equations Calculator, Part 1 A quadratic equation is a second degree polynomial having the general form ax^2 + bx + c = 0, where a, b, and c...

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