Solve 8left(8+19right)=(x+4)left(4x+4right)

$$8 \left( 8+19 \right) = (x+4) \left( 4x+4 \right)$$

Solve for x

$$x=-10$$

$$x=5$$

Steps Using the Quadratic Formula

Add 8 and 19 to get 27.

$$8\times 27=\left(x+4\right)\left(4x+4\right)$$

Multiply 8 and 27 to get 216.

$$216=\left(x+4\right)\left(4x+4\right)$$

Use the distributive property to multiply x+4 by 4x+4 and combine like terms.

$$216=4x^{2}+20x+16$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+20x+16=216$$

Subtract 216 from both sides.

$$4x^{2}+20x+16-216=0$$

Subtract 216 from 16 to get -200.

$$4x^{2}+20x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 20 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 4\left(-200\right)}}{2\times 4}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 4\left(-200\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-20±\sqrt{400-16\left(-200\right)}}{2\times 4}$$

Multiply -16 times -200.

$$x=\frac{-20±\sqrt{400+3200}}{2\times 4}$$

Add 400 to 3200.

$$x=\frac{-20±\sqrt{3600}}{2\times 4}$$

Take the square root of 3600.

$$x=\frac{-20±60}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-20±60}{8}$$

Now solve the equation x=\frac{-20±60}{8} when ± is plus. Add -20 to 60.

$$x=\frac{40}{8}$$

Divide 40 by 8.

$$x=5$$

Now solve the equation x=\frac{-20±60}{8} when ± is minus. Subtract 60 from -20.

$$x=-\frac{80}{8}$$

Divide -80 by 8.

$$x=-10$$

The equation is now solved.

$$x=5 x=-10$$

Steps for Completing the Square

Add 8 and 19 to get 27.

$$8\times 27=\left(x+4\right)\left(4x+4\right)$$

Multiply 8 and 27 to get 216.

$$216=\left(x+4\right)\left(4x+4\right)$$

Use the distributive property to multiply x+4 by 4x+4 and combine like terms.

$$216=4x^{2}+20x+16$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+20x+16=216$$

Subtract 16 from both sides.

$$4x^{2}+20x=216-16$$

Subtract 16 from 216 to get 200.

$$4x^{2}+20x=200$$

Divide both sides by 4.

$$\frac{4x^{2}+20x}{4}=\frac{200}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{20}{4}x=\frac{200}{4}$$

Divide 20 by 4.

$$x^{2}+5x=\frac{200}{4}$$

Divide 200 by 4.

$$x^{2}+5x=50$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=50+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=50+\frac{25}{4}$$

Add 50 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{225}{4}$$

Factor x^{2}+5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{15}{2} x+\frac{5}{2}=-\frac{15}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=5 x=-10$$