Solve (x+12)^2=625

$$( x + 12 ) ^ { 2 } = 625$$

Solve for x

$$x=13$$

$$x=-37$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$x^{2}+24x+144=625$$

Subtract 625 from both sides.

$$x^{2}+24x+144-625=0$$

Subtract 625 from 144 to get -481.

$$x^{2}+24x-481=0$$

To solve the equation, factor x^{2}+24x-481 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=24 ab=-481$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -481.

$$-1,481 -13,37$$

Calculate the sum for each pair.

$$-1+481=480 -13+37=24$$

The solution is the pair that gives sum 24.

$$a=-13 b=37$$

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

$$\left(x-13\right)\left(x+37\right)$$

To find equation solutions, solve x-13=0 and x+37=0.

$$x=13 x=-37$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$x^{2}+24x+144=625$$

Subtract 625 from both sides.

$$x^{2}+24x+144-625=0$$

Subtract 625 from 144 to get -481.

$$x^{2}+24x-481=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-481. To find a and b, set up a system to be solved.

$$a+b=24 ab=1\left(-481\right)=-481$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -481.

$$-1,481 -13,37$$

Calculate the sum for each pair.

$$-1+481=480 -13+37=24$$

The solution is the pair that gives sum 24.

$$a=-13 b=37$$

Rewrite x^{2}+24x-481 as \left(x^{2}-13x\right)+\left(37x-481\right).

$$\left(x^{2}-13x\right)+\left(37x-481\right)$$

Factor out x in the first and 37 in the second group.

$$x\left(x-13\right)+37\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x+37\right)$$

To find equation solutions, solve x-13=0 and x+37=0.

$$x=13 x=-37$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+12\right)^{2}.

$$x^{2}+24x+144=625$$

Subtract 625 from both sides.

$$x^{2}+24x+144-625=0$$

Subtract 625 from 144 to get -481.

$$x^{2}+24x-481=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and -481 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-481\right)}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-481\right)}}{2}$$

Multiply -4 times -481.

$$x=\frac{-24±\sqrt{576+1924}}{2}$$

Add 576 to 1924.

$$x=\frac{-24±\sqrt{2500}}{2}$$

Take the square root of 2500.

$$x=\frac{-24±50}{2}$$

Now solve the equation x=\frac{-24±50}{2} when ± is plus. Add -24 to 50.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{-24±50}{2} when ± is minus. Subtract 50 from -24.

$$x=-\frac{74}{2}$$

Divide -74 by 2.

$$x=-37$$

The equation is now solved.

$$x=13 x=-37$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{625}$$

Simplify.

$$x+12=25 x+12=-25$$

Subtract 12 from both sides of the equation.

$$x=13 x=-37$$