Solve (x+7)^2=25

$$( x + 7 ) ^ { 2 } = 25$$

Solve for x

$$x=-2$$

$$x=-12$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=25$$

Subtract 25 from both sides.

$$x^{2}+14x+49-25=0$$

Subtract 25 from 49 to get 24.

$$x^{2}+14x+24=0$$

To solve the equation, factor x^{2}+14x+24 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.

$$a+b=14 ab=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 14.

$$a=2 b=12$$

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

$$\left(x+2\right)\left(x+12\right)$$

To find equation solutions, solve x+2=0 and x+12=0.

$$x=-2 x=-12$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=25$$

Subtract 25 from both sides.

$$x^{2}+14x+49-25=0$$

Subtract 25 from 49 to get 24.

$$x^{2}+14x+24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 24=24$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.

$$1,24 2,12 3,8 4,6$$

Calculate the sum for each pair.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

The solution is the pair that gives sum 14.

$$a=2 b=12$$

Rewrite x^{2}+14x+24 as \left(x^{2}+2x\right)+\left(12x+24\right).

$$\left(x^{2}+2x\right)+\left(12x+24\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x+2\right)+12\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+12\right)$$

To find equation solutions, solve x+2=0 and x+12=0.

$$x=-2 x=-12$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=25$$

Subtract 25 from both sides.

$$x^{2}+14x+49-25=0$$

Subtract 25 from 49 to get 24.

$$x^{2}+14x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 24}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 24}}{2}$$

Multiply -4 times 24.

$$x=\frac{-14±\sqrt{196-96}}{2}$$

Add 196 to -96.

$$x=\frac{-14±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-14±10}{2}$$

Now solve the equation x=\frac{-14±10}{2} when ± is plus. Add -14 to 10.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-14±10}{2} when ± is minus. Subtract 10 from -14.

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

Divide -24 by 2.

$$x=-12$$

The equation is now solved.

$$x=-2 x=-12$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+7=5 x+7=-5$$

Subtract 7 from both sides of the equation.

$$x=-2 x=-12$$