Quadratic Equations Worksheet

Q: What are the ways to solve a quadratic equation?

There are four main methods: taking square roots (when there is no x term), factoring (when the quadratic factors with integers), completing the square, and the quadratic formula. The quadratic formula works for every quadratic equation.

Q: What is the quadratic formula?

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a). Identify a, b and c with their signs, substitute, and simplify. For x² + 3x − 1 = 0, a = 1, b = 3 and c = −1, so x = (−3 ± √13)/2.

Q: Which method should I use?

Try factoring first if the numbers are small, because it is fastest. If there is no x term, take square roots. If the quadratic will not factor, use the quadratic formula. Completing the square is most useful for rewriting a quadratic in vertex form.

Q: What does the discriminant tell you?

The discriminant is b² − 4ac, the part under the square root in the quadratic formula. If it is positive there are two real solutions, if it is zero there is one repeated real solution, and if it is negative there are no real solutions (the parabola never crosses the x-axis).

Q: Why do I need ± when taking a square root?

Because a positive and a negative number both square to the same value. If x² = 49, then x = 7 or x = −7, since 7² = 49 and (−7)² = 49. Leaving out the negative answer is one of the most common mistakes.

29 printable problems covering every way to solve a quadratic: square roots, factoring, completing the square, the quadratic formula and the discriminant, plus word problems. The answer key shows the key step for each one.

29 problems · answer key included · free, no sign-up

Quadratic Equations Worksheet

learnappu.com/math/algebra/quadratic-equations/worksheet

Name ______________________Date __________ Period ____

Teachers and parents: print the student copy for class or homework, or print with the answer key on a separate page. You can also link to this page from your course site or LMS; no account is needed. Problems get harder within each section.

A. Solve by taking square roots

Isolate the squared term, then take the square root of both sides. Remember ±.

1
$x^{2} = 49$
2
$x^{2} - 25 = 0$
3
$3 x^{2} = 48$
4
$(x - 2)^{2} = 9$
5
$2 x^{2} + 1 = 19$

B. Solve by factoring

Set the equation equal to 0, factor, and set each factor equal to 0.

6
$x^{2} + 5 x + 6 = 0$
7
$x^{2} - 7 x + 12 = 0$
8
$x^{2} + 2 x - 15 = 0$
9
$x^{2} - 9 x = 0$
10
$x^{2} - 16 = 0$
11
$2 x^{2} + 7 x + 3 = 0$
12
$3 x^{2} - 10 x - 8 = 0$
13
$x^{2} + 6 x + 9 = 0$

C. Completing the square

Move the constant, add (b/2)² to both sides, then take square roots.

14
$x^{2} + 6 x + 5 = 0$
15
$x^{2} - 4 x - 1 = 0$
16
$x^{2} + 8 x = 9$
17
$x^{2} - 2 x - 7 = 0$

D. Quadratic formula

Identify a, b and c, then use the formula. Simplify the answer.

18
$x^{2} + 3 x - 1 = 0$
19
$2 x^{2} - 5 x + 1 = 0$
20
$3 x^{2} + x - 2 = 0$
21
$x^{2} - 6 x + 10 = 0$
22
$4 x^{2} - 12 x + 9 = 0$

E. The discriminant

Find b² − 4ac and state how many real solutions the equation has.

23
$x^{2} + 4 x + 4 = 0$
24
$x^{2} - 3 x + 5 = 0$
25
$2 x^{2} + x - 6 = 0$
26
$3 x^{2} - 6 x + 2 = 0$

F. Word problems

Write a quadratic equation, solve it, and check that your answer makes sense.

27
A ball is thrown upward. Its height after $t$ seconds is $h = - 16 t^{2} + 64 t$ feet. When does it hit the ground?
28
A rectangle's length is 4 m more than its width, and its area is 60 m². Find the width and length.
29
The product of two consecutive positive integers is 132. Find the integers.

Answer key

Check your work after you finish each section.

A. Solve by taking square roots

1.
$x = \pm 7$
Take the square root of both sides
2.
$x = \pm 5$
Add 25, then square root
3.
$x = \pm 4$
Divide by 3: $x^{2} = 16$
4.
$x = 5, x = - 1$
$x - 2 = \pm 3$
5.
$x = \pm 3$
Subtract 1, divide by 2: $x^{2} = 9$

B. Solve by factoring

6.
$x = - 2, x = - 3$
$(x + 2) (x + 3) = 0$
7.
$x = 3, x = 4$
$(x - 3) (x - 4) = 0$
8.
$x = 3, x = - 5$
$(x + 5) (x - 3) = 0$
9.
$x = 0, x = 9$
$x (x - 9) = 0$
10.
$x = \pm 4$
Difference of squares: $(x + 4) (x - 4) = 0$
11.
$x = - 3, x = - \frac{1}{2}$
$(2 x + 1) (x + 3) = 0$
12.
$x = 4, x = - \frac{2}{3}$
$(3 x + 2) (x - 4) = 0$
13.
$x = - 3$
Perfect square: $(x + 3)^{2} = 0$ , one repeated root

C. Completing the square

14.
$x = - 1, x = - 5$
$(x + 3)^{2} = 4$ , so $x + 3 = \pm 2$
15.
$x = 2 \pm 5$
$(x - 2)^{2} = 5$
16.
$x = 1, x = - 9$
$(x + 4)^{2} = 25$ , so $x + 4 = \pm 5$
17.
$x = 1 \pm 22$
$(x - 1)^{2} = 8$ , and $8 = 22$

D. Quadratic formula

18.
$x = \frac{- 3 \pm 13}{2}$
$a = 1, b = 3, c = - 1$ ; $b^{2} - 4 a c = 13$
19.
$x = \frac{5 \pm 17}{4}$
$a = 2, b = - 5, c = 1$ ; $b^{2} - 4 a c = 17$
20.
$x = \frac{2}{3}, x = - 1$
$b^{2} - 4 a c = 25$ , so $x = \frac{- 1 \pm 5}{6}$
21.
$No real solutions$
$b^{2} - 4 a c = 36 - 40 = - 4 < 0$
22.
$x = \frac{3}{2}$
$b^{2} - 4 a c = 0$ : one repeated root

E. The discriminant

23.
$b^{2} - 4 a c = 0$
Zero, so one real solution (a repeated root)
24.
$b^{2} - 4 a c = - 11$
Negative, so no real solutions
25.
$b^{2} - 4 a c = 49$
Positive, so two real solutions
26.
$b^{2} - 4 a c = 12$
Positive, so two real solutions

F. Word problems

27.
$t = 4$ seconds ( $t = 0$ is the launch)
$- 16 t^{2} + 64 t = 0$ , so $- 16 t (t - 4) = 0$
28.
Width $6$ m, length $10$ m (reject $w = - 10$ )
$w (w + 4) = 60$ , so $(w + 10) (w - 6) = 0$
29.
$11$ and $12$
$n (n + 1) = 132$ , so $(n + 12) (n - 11) = 0$

How to solve quadratic equations

1. Write it in standard form, $a x^{2} + b x + c = 0$ . Everything must be on one side, with 0 on the other.

2. Try factoring. For $x^{2} + 5 x + 6 = 0$ , find two numbers that multiply to 6 and add to 5: 2 and 3. So $(x + 2) (x + 3) = 0$ , and $x = - 2$ or $x = - 3$ .

3. If it will not factor, use the quadratic formula:

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

4. Check the discriminant $b^{2} - 4 a c$ first if you only need to know how many real solutions there are: positive means two, zero means one, negative means none.

5. Check each answer by substituting it back into the original equation.

Read the full Quadratic Equations lesson with worked examples →

Frequently asked questions

What are the ways to solve a quadratic equation?

There are four main methods: taking square roots (when there is no x term), factoring (when the quadratic factors with integers), completing the square, and the quadratic formula. The quadratic formula works for every quadratic equation.

What is the quadratic formula?

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a). Identify a, b and c with their signs, substitute, and simplify. For x² + 3x − 1 = 0, a = 1, b = 3 and c = −1, so x = (−3 ± √13)/2.

Which method should I use?

Try factoring first if the numbers are small, because it is fastest. If there is no x term, take square roots. If the quadratic will not factor, use the quadratic formula. Completing the square is most useful for rewriting a quadratic in vertex form.

What does the discriminant tell you?

The discriminant is b² − 4ac, the part under the square root in the quadratic formula. If it is positive there are two real solutions, if it is zero there is one repeated real solution, and if it is negative there are no real solutions (the parabola never crosses the x-axis).

Why do I need ± when taking a square root?

Because a positive and a negative number both square to the same value. If x² = 49, then x = 7 or x = −7, since 7² = 49 and (−7)² = 49. Leaving out the negative answer is one of the most common mistakes.

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