Solve k^2-4(8k+36)=0

$$k ^ { 2 } - 4 ( 8 k + 36 ) = 0$$

Solve for k

$$k=-4$$

$$k=36$$

Steps Using Factoring

Use the distributive property to multiply -4 by 8k+36.

$$k^{2}-32k-144=0$$

To solve the equation, factor k^{2}-32k-144 using formula k^{2}+\left(a+b\right)k+ab=\left(k+a\right)\left(k+b\right). To find a and b, set up a system to be solved.

$$a+b=-32 ab=-144$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Calculate the sum for each pair.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

The solution is the pair that gives sum -32.

$$a=-36 b=4$$

Rewrite factored expression \left(k+a\right)\left(k+b\right) using the obtained values.

$$\left(k-36\right)\left(k+4\right)$$

To find equation solutions, solve k-36=0 and k+4=0.

$$k=36 k=-4$$

Steps Using Factoring By Grouping

Use the distributive property to multiply -4 by 8k+36.

$$k^{2}-32k-144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as k^{2}+ak+bk-144. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\left(-144\right)=-144$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Calculate the sum for each pair.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

The solution is the pair that gives sum -32.

$$a=-36 b=4$$

Rewrite k^{2}-32k-144 as \left(k^{2}-36k\right)+\left(4k-144\right).

$$\left(k^{2}-36k\right)+\left(4k-144\right)$$

Factor out k in the first and 4 in the second group.

$$k\left(k-36\right)+4\left(k-36\right)$$

Factor out common term k-36 by using distributive property.

$$\left(k-36\right)\left(k+4\right)$$

To find equation solutions, solve k-36=0 and k+4=0.

$$k=36 k=-4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -4 by 8k+36.

$$k^{2}-32k-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -32 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-144\right)}}{2}$$

Square -32.

$$k=\frac{-\left(-32\right)±\sqrt{1024-4\left(-144\right)}}{2}$$

Multiply -4 times -144.

$$k=\frac{-\left(-32\right)±\sqrt{1024+576}}{2}$$

Add 1024 to 576.

$$k=\frac{-\left(-32\right)±\sqrt{1600}}{2}$$

Take the square root of 1600.

$$k=\frac{-\left(-32\right)±40}{2}$$

The opposite of -32 is 32.

$$k=\frac{32±40}{2}$$

Now solve the equation k=\frac{32±40}{2} when ± is plus. Add 32 to 40.

$$k=\frac{72}{2}$$

Divide 72 by 2.

$$k=36$$

Now solve the equation k=\frac{32±40}{2} when ± is minus. Subtract 40 from 32.

$$k=-\frac{8}{2}$$

Divide -8 by 2.

$$k=-4$$

The equation is now solved.

$$k=36 k=-4$$

Steps for Completing the Square

Use the distributive property to multiply -4 by 8k+36.

$$k^{2}-32k-144=0$$

Add 144 to both sides. Anything plus zero gives itself.

$$k^{2}-32k=144$$

Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-32k+\left(-16\right)^{2}=144+\left(-16\right)^{2}$$

Square -16.

$$k^{2}-32k+256=144+256$$

Add 144 to 256.

$$k^{2}-32k+256=400$$

Factor k^{2}-32k+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-16\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-16\right)^{2}}=\sqrt{400}$$

Simplify.

$$k-16=20 k-16=-20$$

Add 16 to both sides of the equation.

$$k=36 k=-4$$