Introduction
This set of practice questions will give you practice in finding solutions to quadratic equations. Quadratic equations are those equations that can be written in the form ax^{2} + bx + c = 0, where a ≠ 0. While there are several methods for solving quadratic equations, solutions for all the equations presented here can be found by factoring.
In some algebra problems, you practiced factoring polynomials by using the greatest common factor method, the difference of two perfect squares method, and the trinomial factor method. Use these methods to factor the equations that have been transformed into quadratic equations. Then, using the zero product property (if (a)(b) = 0, then a = 0 or b = 0 or both = 0), let each factor equal zero and solve for the variable. There will be two solutions for each quadratic equation. (Ignore numerical factors such as the 3 in the factored equation 3(x + 1)(x + 1) = 0 when finding solutions to quadratic equations. The solutions will be the same for equations with or without the numerical factors.)
Practice Questions
Find the solutions to the following quadratic equations.
- x^{2} – 25 = 0
- n^{2} – 169 = 0
- a^{2} + 12a + 32 = 0
- y^{2} – 15y + 56 = 0
- b^{2} + b – 90 = 0
- 4x^{2} = 49
- 25r^{2} = 144
- 2n^{2} + 20n + 42 = 0
- 3c^{2} – 33c – 78 = 0
- 100r^{2} = 144
- 3x^{2} – 36x + 108 = 0
- 7a^{2} – 21a – 28 = 0
- 8y^{2} + 56y + 96 = 0
- 2x^{2} + 9x = –10
- 4x^{2} + 4x = 15
- 9x^{2} + 12x = –4
- 3x^{2} = 19x – 20
- 8b^{2} + 10b = 42
- 14n^{2} = 7n + 21
- 6b^{2} + 20b = –9b – 20
- 15x^{2} – 70x – 120 = 0
- 7x^{2} = 52x – 21
- 36z^{2} + 78z = –36
- 12r^{2} = 192 – 40r
- 24x^{2} = 3(43x – 15)
Answers
Numerical expressions in parentheses like this [ ] are operations performed on only part of the original expression. The operations performed within these symbols are intended to show how to evaluate the various terms that make up the entire expression.
Expressions with parentheses that look like this () contain either numerical substitutions or expressions that are part of a numerical expression. Once a single number appears within these parentheses, the parentheses are no longer needed and need not be used the next time the entire expression is written.
When two pair of parentheses appear side by side like this ()(), it means that the expressions within are to be multiplied.
Sometimes parentheses appear within other parentheses in numerical or algebraic expressions. Regardless of what symbol is used, (), { }, or [ ], perform operations in the innermost parentheses first and work outward.
The solutions are underlined.
1. The expression is the difference of two perfect squares. | |
The equation factors into | (x + 5)(x – 5) = 0. |
Applying the zero product property (if (a)(b) = 0, then a = 0 or b = 0 or both = 0), the first factor or the second factor or both must equal zero. | (x + 5) = 0 |
Subtract 5 from both sides of the equation. | x + 5 – 5 = 0 – 5 |
Combine like terms on each side. | x = –5 |
Let the second factor equal zero. | x – 5 = 0 |
Add 5 to both sides of the equation. | x – 5 + 5 = 0 + 5 |
Combine like terms on each side. | x = 5 |
The solutions for the equation are x = 5 and x = –5. | |
2. The expression is the difference of two perfect squares. | |
The equation factors into | (n + 13)(n – 13) = 0. |
Applying the zero product property (if (a)(b) = 0, then a = 0 or b = 0 or both = 0), the first factor or the second factor or both must equal zero. | (n + 13) = 0 |
Subtract 13 from both sides of the equation. | n + 13 – 13 = 0 – 13 |
Combine like terms on each side. | n = –13 |
Let the second factor equal zero. | n – 13 = 0 |
Add 13 to both sides of the equation. | n – 13 + 13 = 0 + 13 |
Combine like terms on each side. | n = 13 |
The solutions for the equation are n = 13 and n = –13. | |
3. Factor the trinomial expression using the trinomial factor form. | (a + 4)(a + 8) = 0 |
Using the zero product property, subtract 4 from both sides. | (a + 4) = 0 |
a + 4 – 4 = 0 – 4 | |
Combine like terms on each side. | a = –4 |
Let the second factor equal zero. | (a + 8) = 0 |
Subtract 8 from both sides. | a + 8 – 8 = 0 – 8 |
Combine like terms on each side. | a = –8 |
. | |
4. Factor the trinomial expression using the trinomial factor form. | (y – 8)(y – 7) = 0 |
Using the zero product property, add 8 to both sides. | (y – 8) = 0 |
y – 8 + 8 = 0 + 8 | |
Combine like terms on each side. | y = 8 |
Let the second factor equal zero. | (y – 7) = 0 |
Add 7 to both sides. | y – 7 + 7 = 0 + 7 |
Combine like terms on each side. | y = 7 |
. | |
5. Factor the trinomial expression using the trinomial factor form. | (b + 10)(b – 9) = 0 |
Using the zero product property, subtract 10 from both sides. | (b + 10) = 0 |
b + 10 – 10 = 0 – 10 | |
Combine like terms on each side. | b = –10 |
Let the second factor equal zero. | (b – 9) = 0 |
Add 9 to both sides. | b – 9 + 9 = 0 + 9 |
Combine like terms on each side. | b = 9 |
. | |
6. Transform the equation so that all terms are on one side and are equal to zero. | |
Subtract 49 from both sides. | 4x^{2} – 49 = 49 – 49 |
Combine like terms on each side. | 4x^{2} – 49 = 0 |
The expression is the difference of two perfect squares. | |
The equation factors into | (2x + 7)(2x – 7) = 0. |
Applying the zero product property (if (a)(b) = 0, then a = 0 or b = 0 or both = 0), the first factor or the second factor or both must equal zero. | (2x + 7) = 0 |
Subtract 7 from both sides of the equation. | 2x + 7 – 7 = 0 – 7 |
Combine like terms on each side. | 2x = –7 |
Divide both sides by 2. | |
Simplify. | x = – |
Let the second factor equal zero. | (2x – 7) = 0 |
Add 7 to both sides of the equation. | 2x – 7 + 7 = 0 + 7 |
Combine like terms on both sides. | 2x = 7 |
Divide both sides by 2. | x = |
7. Transform the equation so that all terms are on one side and are equal to zero. | |
Subtract 144 from both sides. | 25r^{2} – 144 = 144 – 144 |
Combine like terms on each side. | 25r^{2} – 144 = 0 |
The expression is the difference of two perfect squares. | (5r + 12)(5r – 12) = 0. |
Applying the zero product property (if (a)(b) = 0, then a = 0 or b = 0 or both = 0), the first factor or the second factor or both must equal zero. | (5r + 12) = 0 |
Subtract 12 from both sides of the equation. | 5r + 12 – 12 = 0 – 12 |
Combine like terms on each side. | 5r = –12 |
Divide both sides by 5. | |
Simplify. | r = – |
Let the second factor equal zero. | (5r – 12) = 0 |
Add 12 to both sides of the equation. | 5r – 12 + 12 = 0 + 12 |
Combine like terms on each side. | 5r = 12 |
Divide both sides by 5. | |
Simplify. | r = |
8. Factor the trinomial expression using the trinomial factor form. | (2n + 6)(n + 7) = 0 |
Using the zero product property, subtract 6 from both sides. | (2n + 6) = 0 |
2n + 6 – 6 = 0 – 6 | |
Combine like terms on each side. | 2n = –6 |
Divide both sides by 2. | |
Simplify terms. | n = ^{–}3 |
Let the second factor equal zero. | (n + 7) = 0 |
Subtract 7 from both sides. | n + 7 – 7 = 0 – 7 |
Combine like terms on each side. | n = –7 |
9. Use the greatest common factor method. | 3(c^{2} – 11c – 26) = 0 |
Factor the trinomial expression using the trinomial factor form. | 3(c – 13)(c + 2) = 0 |
Ignore the factor 3 in the expression. | |
Using the zero product property, add 13 to both sides. | (c – 13) = 0 |
c – 13 + 13 = 0 + 13 | |
Combine like terms on each side. | c = 13 |
Let the second factor equal zero. | (c + 2) = 0 |
Subtract 2 from both sides. | c + 2 – 2 = 0 – 2 |
Combine like terms on each side. | c = –2 |
10. Transform the equation so that all terms are on one side and are equal to zero. | |
Subtract 144 from both sides. | 100r^{2} – 144 = 144 – 144 |
Combine like terms on each side. | 100r^{2} – 144 = 0 |
The expression is the difference of two perfect squares. | |
The equation factors into | (10r + 12)(10r – 12) = 0. |
Applying the zero product property (if (a)(b) = 0, then a = 0 or b = 0 or both = 0), the first factor or the second factor or both must equal zero. | (10r + 12) = 0 |
Subtract 12 from both sides of the equation. | 10r + 12 – 12 = 0 – 12 |
Combine like terms on each side. | 10r = –12 |
Divide both sides by 10. | |
Simplify terms. | r = – |
Let the second factor equal zero. | (10r – 12) = 0 |
Add 12 to both sides of the equation. | 10r – 12 + 12 = 0 + 12 |
Combine like terms on each side. | 10r = 12 |
Divide both sides by 10. | |
Simplify terms. | r = |
11. Use the greatest common factor method. | 3(x^{2} – 12x + 36) = 0 |
Factor the trinomial expression using the trinomial factor form. | 3(x – 6)(x – 6) = 0 |
Ignore the factor 3 in the expression. | |
Using the zero product property, add 6 to both sides. | (x – 6) = 0 |
x – 6 + 6 = 0 + 6 | |
Combine like terms on each side. | x = 6 |
12. Use the greatest common factor method. | 7(a^{2} – 3a – 4) = 0 |
Factor the trinomial expression using the trinomial factor form. | 7(a – 4)(a + 1) = 0 |
Ignore the factor 7 in the expression. | |
Using the zero product property, add 4 to both sides. | (a – 4) = 0 |
a – 4 + 4 = 0 + 4 | |
Combine like terms on each side. | a = 4 |
Let the second factor equal zero. | (a + 1) = 0 |
Subtract 2 from both sides. | a + 1 – 1 = 0 – 1 |
Combine like terms on each side. | a = –1 |
The solutions for the quadratic equation 7a2 – 21a – 70 = 0 are a = –2 and a = 5. | |
13. Use the greatest common factor method. | 8(y^{2} + 7y + 12) = 0 |
Factor the trinomial expression using the trinomial factor form. | 8(y + 4)(y + 3) = 0 |
Ignore the factor 8 in the expression. | |
Using the zero product property, subtract 4 from both sides. | (y + 4) = 0 |
(y + 4) = 0 | |
Combine like terms on each side. | y = –4 |
Let the second factor equal zero. | (y + 3) = 0 |
Subtract 3 from both sides. | y + 3 – 3 = 0 – 3 |
Simplify. | y = –3 |
The solutions for the quadratic equation 8y^{2} + 56y + 96 = 0 are y = –4 and y = –3. | |
14. Transform the equation into the familiar trinomial equation form. | |
Add 10 from both sides of the equation. | 2x^{2} + 9x + 10 = –10 + 10 |
Combine like terms on each side. | 2x^{2} + 9x + 10 = 0 |
Factor the trinomial expression using the trinomial factor form. | (2x + 5)(x + 2) = 0 |
Using the zero product property, subtract 5 from both sides. | (2x + 5) = 0 |
2x + 5 – 5 = 0 – 5 | |
Combine like terms on each side. | 2x = –5 |
Divide both sides by 2. | |
Simplify terms. | |
Let the second term equal zero. | x + 2 = 0 |
Subtract 2 to both sides. | x + 2 – 2 = 0 – 2 |
Simplify. | x = –2 |
15. Transform the equation into the familiar trinomial equation form. | |
Subtract 15 from both sides of the equation. | 4x^{2} + 4x – 15 = 15 – 15 |
Combine like terms on each side. | 4x^{2} + 4x – 15 = 0 |
Factor the trinomial expression using the trinomial factor form. | (2x – 3)(2x + 5) = 0 |
Using the zero product property, subtract from both sides. | 5 (2x + 5) = 0 |
2x + 5 – 5 = 0 – 5 | |
Simplify. | 2x = –5 |
Divide both sides by 2. | |
Simplify terms. | |
Let the second factor equal zero. | (2x – 3) = 0 |
Add 3 to both sides. | 2x – 3 + 3 = 0 + 3 |
Simplify. | 2x = 3 |
Divide both sides by 2. | |
Simplify terms. | |
16. Transform the equation into the familiar trinomial equation form. | |
Add 4 to both sides of the equation. | 9x^{2} + 12x + 4 = –4 + 4 |
Combine like terms on each side. | 9x^{2} + 12x + 4 = 0 |
Factor the trinomial expression using the trinomial factor form. | (3x + 2)(3x + 2) = 0 |
Using the zero product property, subtract 2 from both sides. | (3x + 2) = 0 |
(3x + 2) = 0 | |
Simplify. | 3x = –2 |
Divide both sides by 3. | |
Simplify terms. | |
17. Transform the equation into the familiar trinomial equation form. | |
Subtract 19x from both sides. | 3x^{2} – 19x = 19x – 19x – 20 |
Combine like terms. | 3x^{2} – 19x = –20 |
Add 20 to both sides. | 3x^{2} – 19x + 20 = –20 + 20 |
Combine like terms on each side. | 3x^{2} – 19x + 20 = 0 |
Factor the trinomial expression using the trinomial factor form. | (3x – 4)(x – 5) = 0 |
Using the zero product property, add to both sides. | 4 (3x – 4) = 0 |
3x – 4 + 4 = 0 + 4 | |
Simplify. | 3x = 4 |
Divide both sides by 3. | |
Simplify terms. | |
Now let the second term equal zero. | x – 5 = 0 |
Add 5 to both sides. | x – 5 + 5 = 0 + 5 |
Simplify. | x = 5 |
18. Transform the equation into the familiar trinomial equation form. | |
Subtract 42 from both sides of the equation. | 8b^{2} + 10b – 42 = 42 – 42 |
Simplify. | 8b^{2} + 10b – 42 = 0 |
Use the greatest common factor method to factor out 2. | 2(4b^{2} + 5b – 21) = 0 |
Factor the trinomial expression using the trinomial factor form. | 2(4b – 7)(b + 3) = 0 |
Ignore the factor 2 in the expression. | |
Using the zero product property, add 7 to both sides. | (4b – 7) = 0 |
4b – 7 + 7 = 0 + 7 | |
Simplify. | 4b = 7 |
Divide both sides by 4. | |
Simplify terms. | |
Now let the second term equal zero. | (b + 3) = 0 |
Subtract 3 from both sides. | b + 3 – 3 = 0 – 3 |
Simplify. | b = –3 |
19. Transform the equation into the familiar trinomial equation form. | |
Subtract 7n from both sides of the equation. | 14n^{2} – 7n = 7n – 7n + 21 |
Simplify and subtract 21 from both sides. | 14n^{2} – 7n – 21 = 21 – 21 |
Simplify the equation. | 14n^{2} – 7n – 21 = 0 |
Factor the greatest common factor from each term. | 7(2n^{2} – n – 3) = 0 |
Factor the trinomial expression using the trinomial factor form. | 7(2n – 3)(n + 1) = 0 |
Ignore the factor 7 in the expression. | |
Using the zero product property, add 3 to both sides. | (2n – 3) = 0 |
2n – 3 + 3 = 0 + 3 | |
Simplify. | 2n = 3 |
Divide both sides by 2. | |
Simplify terms. | |
Now set the second equal to zero. | n + 1 = 0 |
Subtract 1 from both sides. | n + 1 – 1 = 0 – 1 |
Simplify. | n = –1 |
20. Transform the equation into the familiar trinomial equation form. | |
Add 9b to both sides of the equation. | 6b^{2} + 20b + 9b = ^{–}9b + 9b – 20 |
Simplify and add 20 to both sides of the equation. | 6b^{2} + 29b + 20 = 20 – 20 |
Simplify. | 6b^{2} + 29b + 20 = 0 |
Factor the trinomial expression using the trinomial factor form. | (6b + 5)(b + 4) = 0 |
Using the zero product property, subtract 5 from both sides. | (6b + 5) = 0 |
6b + 5 – 5 = 0 – 5 | |
Simplify. | 6b = ^{–}5 |
Divide both sides by 6. | |
Simplify terms. | |
Now set the second factor equal to zero. | b + 4 = 0 |
Subtract 4 from both sides. | b + 4 – 4 = 0 – 4 |
Simplify. | b = ^{–}4 |
21. Factor the greatest common factor from each term. | 5(3x^{2} – 14x – 24) = 0 |
Now factor the trinomial expression using the trinomial factor form. | 5(3x + 4)(x – 6) = 0 |
Using the zero product property, subtract 4 from both sides. | (3x + 4) = 0 |
3x + 4 – 4 = 0 – 4 | |
Simplify. | 3x = –4 |
Divide both sides by 3. | |
Simplify terms. | |
Add 6 to both sides. | x – 6 + 6 = 0 + 6 |
Simplify. | x = 6 |
22. Transform the equation into the familiar trinomial equation form. | |
Subtract 52x from both sides of the equation. | 7x^{2} – 52x = 52x – 52x – 21 |
Simplify and add 21 to both sides of the equation. | 7x^{2} – 52x + 21 = 21 – 21 |
Simplify. | 7x^{2} – 52x + 21 = 0 |
Factor the trinomial expression using the trinomial factor form. | (7x – 3)(x – 7) = 0 |
Using the zero product property, add 3 to both sides. | (7x – 3) = 0 |
7x – 3 + 3 = 0 + 3 | |
Simplify. | 7x = 3 |
Divide both sides by 7. | |
Simplify terms. | |
Now set the second factor equal to zero. | x – 7 = 0 |
Add 7 to both sides of the equation. | x – 7 + 7 = 0 + 7 |
Simplify. | x = 7 |
23. Transform the equation into the familiar trinomial equation form. | |
Add 36 to both sides of the equation. | 36z^{2} + 78z + 36 = –36 + 36 |
Combine like terms. | 36z_{2} + 78z + 36 = 0 |
Factor out the greatest common factor from each term. | 6(6z^{2} + 13z + 6) = 0 |
Factor the trinomial expression into two factors. | 6(2z + 3)(3z + 2) = 0 |
Ignore the numerical factor and set the first factor equal to zero. | (2z + 3) = 0 |
Subtract 3 from both sides. | 2z + 3 – 3 = 0 – 3 |
Simplify terms. | 2z = –3 |
Divide both sides by 2. | |
Simplify terms. | |
Now let the second factor equal zero. | (3z + 2) = 0 |
Subtract 2 from both sides. | 3z + 2 – 2 = 0 – 2 |
Simplify terms. | 3z = –2 |
Divide both sides by 3. | |
Simplify terms. | |
24.Transform the equation into the familiar trinomial equation form. | |
Add (40r – 192) to both sides of the equation. | 12r^{2} + 40r – 192 = 192 – 40r + 40r – 192 |
Combine like terms. | 12r^{2} + 40r – 192 = 0 |
Factor the greatest common factor, 4, out of each term. | 4(3r^{2} + 10r – 48) = 0 |
Now factor the trinomial expression. | 4(r + 6)(3r – 8) = 0 |
Ignoring the numerical factor, set one factor equal to zero. | (r + 6) = 0 |
Subtract 6 from both sides. | r + 6 – 6 = 0 – 6 |
Simplify. | r = –6 |
Now set the second factor equal to zero. | 3r – 8 = 0 |
Add 8 to both sides. | 3r – 8 + 8 = 0 + 8 |
Simplify. | 3r = 8 |
Divide both sides by 3. | |
Simplify terms. | |
25.Divide both sides of the equation by 3. | |
Simplify terms. | 8x^{2} = 43x – 15 |
Add (15 – 43x) to both sides of the equation. | 8x^{2} + 15 – 43x = 43x – 15 + 15 – 43x |
Combine like terms. | 8x^{2} + 15 – 43x = 0 |
Use the commutative property to move terms. | 8x^{2} – 43x + 15 = 0 |
Factor the trinomial expression. | (8x – 3)(x – 5) = 0 |
Using the zero product property, add 3 to both sides and divide by 8. | 8x – 3 = 0 |
Simplify terms. | |
Now let the second factor equal zero. | x – 5 = 0 |
Add 5 to both sides. | x = 5 |
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