Solve 10(10+8)=x(3+x)

$$10 ( 10 + 8 ) = x ( 3 + x )$$

Solve for x

$$x=-15$$

$$x=12$$

Steps Using the Quadratic Formula

Add 10 and 8 to get 18.

$$10\times 18=x\left(3+x\right)$$

Multiply 10 and 18 to get 180.

$$180=x\left(3+x\right)$$

Use the distributive property to multiply x by 3+x.

$$180=3x+x^{2}$$

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

$$3x+x^{2}=180$$

Subtract 180 from both sides.

$$3x+x^{2}-180=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+3x-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-180\right)}}{2}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-180\right)}}{2}$$

Multiply -4 times -180.

$$x=\frac{-3±\sqrt{9+720}}{2}$$

Add 9 to 720.

$$x=\frac{-3±\sqrt{729}}{2}$$

Take the square root of 729.

$$x=\frac{-3±27}{2}$$

Now solve the equation x=\frac{-3±27}{2} when ± is plus. Add -3 to 27.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{-3±27}{2} when ± is minus. Subtract 27 from -3.

$$x=-\frac{30}{2}$$

Divide -30 by 2.

$$x=-15$$

The equation is now solved.

$$x=12 x=-15$$

Steps for Completing the Square

Add 10 and 8 to get 18.

$$10\times 18=x\left(3+x\right)$$

Multiply 10 and 18 to get 180.

$$180=x\left(3+x\right)$$

Use the distributive property to multiply x by 3+x.

$$180=3x+x^{2}$$

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

$$3x+x^{2}=180$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+3x=180$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=180+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=180+\frac{9}{4}$$

Add 180 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{729}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{729}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{27}{2} x+\frac{3}{2}=-\frac{27}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=12 x=-15$$