7.6 Quadratic Equations

Before you get started, take this readiness quiz.

Solve: $5 y - 3 = 0$.
If you missed this problem, review the reference.

Solve: $10 a = 0$.
If you missed this problem, review the reference.

Combine like terms: $12 x^{2} - 6 x + 4 x$.
If you missed this problem, review the reference.

Factor $n^{3} - 9 n^{2} - 22 n$ completely.
If you missed this problem, review the reference.

Solve: $\left(x - 3\right) \left(x + 5\right) = 0$.

Solve: $\left(y - 6\right) \left(y + 9\right) = 0$.

Solve: $\left(3 m - 2\right) \left(2 m + 1\right) = 0$.

Solve: $\left(4 p + 3\right) \left(4 p - 3\right) = 0$.

Solve: $2 u \left(5 u - 1\right) = 0$.

Solve: $w \left(2 w + 3\right) = 0$.

Solve: $\left(x + 1\right)^{2} = 0$.

Solve: $\left(v - 2\right)^{2} = 0$.

Solve: $x^{2} - x - 12 = 0$.

Solve: $b^{2} + 9 b + 14 = 0$.

Solve: $3 c^{2} = 10 c - 8$.

Solve: $2 d^{2} - 5 d = 3$.

Solve: $6 a^{2} + 9 a = 3 a$.

Solve: $45 b^{2} - 2 b = −17 b$.

Solve: $25 p^{2} = 49$.

Solve: $36 x^{2} = 121$.

Solve: $\left(3 x - 8\right) \left(x - 1\right) = 3 x$.

Solve: $\left(2 m + 1\right) \left(m + 3\right) = 12 m$.

Solve: $\left(k + 1\right) \left(k - 1\right) = 8$.

Solve: $8 x^{3} = 24 x^{2} - 18 x$.

Solve: $16 y^{2} = 32 y^{3} + 2 y$.

Solve: $4 x^{2} = 16 x + 84$.

Solve: $18 a^{2} - 30 = −33 a$.

Solve: $123 b = −6 - 60 b^{2}$.

The product of two consecutive integers is 132. Find the integers.

The product of two consecutive integers is $240$. Find the integers.

The product of two consecutive integers is $420$. Find the integers.

A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.

A boat’s sail is a right triangle. The length of one side of the sail is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the sail.

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg.

Solve.

$\left(x - 3\right) \left(x + 7\right) = 0$

Solve.

$\left(3 a - 10\right) \left(2 a - 7\right) = 0$

Solve.

$6 m \left(12 m - 5\right) = 0$

Solve.

$\left(y - 3\right)^{2} = 0$

Solve.

$\left(2 x - 1\right)^{2} = 0$

Solve.

$x^{2} + 7 x + 12 = 0$

Solve.

$5 a^{2} - 26 a = 24$

Solve.

$4 m^{2} = 17 m - 15$

Solve.

$7 a^{2} + 14 a = 7 a$

Solve.

$49 m^{2} = 144$

Solve.

$\left(y - 3\right) \left(y + 2\right) = 4 y$

Solve.

$\left(2 x + 1\right) \left(x - 3\right) = −4 x$

Solve.

$16 p^{3} = 24 p^{2} - 9 p$

Solve.

$20 x^{2} - 60 x = −45$

Solve.

The product of two consecutive integers is 56. Find the integers.

Solve.

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

Solve.

A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

Solve.

$\left(x + 8\right) \left(x - 3\right) = 0$

Solve.

$p^{2} + 12 p + 11 = 0$

Solve.

$m^{2} = 6 m + 16$

Solve.

$a^{3} - a^{2} - 42 a = 0$

Solve.

The product of two consecutive integers is 110. Find the integers.

Area of a patio If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation $\left(s + 4\right)^{2} = 196$ for s to find the length of a side of the patio.

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

Find the greatest common factor.

$42 , 60$

Find the greatest common factor.

$90 , 150 , 105$

Factor the greatest common factor from each polynomial.

$24 x - 42$

Factor the greatest common factor from each polynomial.

$15 m^{4} + 6 m^{2} n$

Factor by grouping.

$a x - a y + b x - b y$

Factor by grouping.

$x^{2} + 7 x - 3 x - 21$

Factor by grouping.

$m^{3} + m^{2} + m + 1$

Factor each trinomial of the form $x^{2} + b x + c$.

$u^{2} + 17 u + 72$

Factor each trinomial of the form $x^{2} + b x + c$.

$k^{2} - 16 k + 60$

Factor each trinomial of the form $x^{2} + b x + c$.

$y^{2} + 6 y - 7$

Factor each trinomial of the form $x^{2} + b x + c$.

$s^{2} - 2 s - 8$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2}$.

$x^{2} + 12 x y + 35 y^{2}$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2}$.

$a^{2} + 4 a b - 21 b^{2}$

Identify the best method to use to factor each polynomial.

$y^{2} - 17 y + 42$

Identify the best method to use to factor each polynomial.

$8 a^{3} + 72 a$

Factor completely.

$6 x^{2} + 42 x + 60$

Factor completely.

$3 n^{4} - 12 n^{3} - 96 n^{2}$

Factor.

$2 x^{2} + 9 x + 4$

Factor.

$18 a^{2} - 9 a + 1$

Factor.

$15 p^{2} + 2 p - 8$

Factor.

$40 s^{2} - s - 6$

Factor.

$3 x^{2} + 3 x - 36$

Factor.

$60 y^{2} - 85 y - 25$

Factor.

$25 x^{2} + 30 x + 9$

Factor.

$36 a^{2} - 84 a b + 49 b^{2}$

Factor.

$40 x^{2} + 360 x + 810$

Factor.

$2 y^{3} - 16 y^{2} + 32 y$

Factor.

$81 r^{2} - 25$

Factor.

$169 m^{2} - n^{2}$

Factor.

$25 p^{2} - 1$

Factor.

$9 - 121 y^{2}$

Factor.

$20 x^{2} - 125$

Factor.

$49 u^{3} - 9 u$

Factor.

$a^{3} - 125$

Factor.

$2 m^{3} + 54$

Factor completely.

$24 x^{3} + 44 x^{2}$

Factor completely.

$16 n^{2} - 56 m n + 49 m^{2}$

Factor completely.

$5 r^{2} + 22 r - 48$

Factor completely.

$n^{4} - 81$

Factor completely.

$5 x^{2} + 5 x - 60$

Factor completely.

$m^{3} + 125$

Solve.

$\left(a - 3\right) \left(a + 7\right) = 0$

Solve.

$3 m \left(2 m - 5\right) \left(m + 6\right) = 0$

Solve.

$x^{2} + 9 x + 20 = 0$

Solve.

$2 p^{2} - 11 p = 40$

Solve.

$144 m^{2} - 25 = 0$

Solve.

The product of two consecutive numbers is $462$. Find the numbers.

Find the Greatest Common Factor in each expression.

$14 y - 42$

Find the Greatest Common Factor in each expression.

$80 a^{2} + 120 a^{3}$

Factor completely.

$x^{2} + 13 x + 42$

Factor completely.

$3 a^{3} - 6 a^{2} - 72 a$

Factor completely.

$5 n^{2} + 30 n + 45$

Factor completely.

$x y - 8 y + 7 x - 56$

Factor completely.

$9 s^{2} - 12 s + 4$

Factor completely.

$100 - a^{2}$

Factor completely.

$3 x^{2} - 75 y^{2}$

Factor completely.

$a b - 3 b - 2 a + 6$

Factor completely.

$8 m^{2} + 22 m + 5$

Solve.

$y^{2} = y + 132$

Solve.

$9 b^{2} - 9 = 0$

Solve.

$4 n^{2} + 19 n + 21 = 0$

Solve.

The product of two consecutive integers is 156. Find the integers.