Use the distributive property to multiply $12+x$ by $18+x$ and combine like terms.
$$216+30x+x^{2}=280$$
Subtract $280$ from both sides.
$$216+30x+x^{2}-280=0$$
Subtract $280$ from $216$ to get $-64$.
$$-64+30x+x^{2}=0$$
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.
$$x^{2}+30x-64=0$$
This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $1$ for $a$, $30$ for $b$, and $-64$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
Now solve the equation $x=\frac{-30±34}{2}$ when $±$ is plus. Add $-30$ to $34$.
$$x=\frac{4}{2}$$
Divide $4$ by $2$.
$$x=2$$
Now solve the equation $x=\frac{-30±34}{2}$ when $±$ is minus. Subtract $34$ from $-30$.
$$x=-\frac{64}{2}$$
Divide $-64$ by $2$.
$$x=-32$$
The equation is now solved.
$$x=2$$ $$x=-32$$
Steps for Completing the Square
Use the distributive property to multiply $12+x$ by $18+x$ and combine like terms.
$$216+30x+x^{2}=280$$
Subtract $216$ from both sides.
$$30x+x^{2}=280-216$$
Subtract $216$ from $280$ to get $64$.
$$30x+x^{2}=64$$
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 $x^{2}+bx=c$.
$$x^{2}+30x=64$$
Divide $30$, the coefficient of the $x$ term, by $2$ to get $15$. Then add the square of $15$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.
$$x^{2}+30x+15^{2}=64+15^{2}$$
Square $15$.
$$x^{2}+30x+225=64+225$$
Add $64$ to $225$.
$$x^{2}+30x+225=289$$
Factor $x^{2}+30x+225$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.
$$\left(x+15\right)^{2}=289$$
Take the square root of both sides of the equation.
