Solve (x+10)^2=25

$$( x + 10 ) ^ { 2 } = 25$$

Solve for x

$$x=-5$$

$$x=-15$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+10\right)^{2}.

$$x^{2}+20x+100=25$$

Subtract 25 from both sides.

$$x^{2}+20x+100-25=0$$

Subtract 25 from 100 to get 75.

$$x^{2}+20x+75=0$$

To solve the equation, factor x^{2}+20x+75 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.

$$a+b=20 ab=75$$

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 75.

$$1,75 3,25 5,15$$

Calculate the sum for each pair.

$$1+75=76 3+25=28 5+15=20$$

The solution is the pair that gives sum 20.

$$a=5 b=15$$

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

$$\left(x+5\right)\left(x+15\right)$$

To find equation solutions, solve x+5=0 and x+15=0.

$$x=-5 x=-15$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+10\right)^{2}.

$$x^{2}+20x+100=25$$

Subtract 25 from both sides.

$$x^{2}+20x+100-25=0$$

Subtract 25 from 100 to get 75.

$$x^{2}+20x+75=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+75. To find a and b, set up a system to be solved.

$$a+b=20 ab=1\times 75=75$$

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 75.

$$1,75 3,25 5,15$$

Calculate the sum for each pair.

$$1+75=76 3+25=28 5+15=20$$

The solution is the pair that gives sum 20.

$$a=5 b=15$$

Rewrite x^{2}+20x+75 as \left(x^{2}+5x\right)+\left(15x+75\right).

$$\left(x^{2}+5x\right)+\left(15x+75\right)$$

Factor out x in the first and 15 in the second group.

$$x\left(x+5\right)+15\left(x+5\right)$$

Factor out common term x+5 by using distributive property.

$$\left(x+5\right)\left(x+15\right)$$

To find equation solutions, solve x+5=0 and x+15=0.

$$x=-5 x=-15$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+10\right)^{2}.

$$x^{2}+20x+100=25$$

Subtract 25 from both sides.

$$x^{2}+20x+100-25=0$$

Subtract 25 from 100 to get 75.

$$x^{2}+20x+75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and 75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\times 75}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 75}}{2}$$

Multiply -4 times 75.

$$x=\frac{-20±\sqrt{400-300}}{2}$$

Add 400 to -300.

$$x=\frac{-20±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-20±10}{2}$$

Now solve the equation x=\frac{-20±10}{2} when ± is plus. Add -20 to 10.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

Now solve the equation x=\frac{-20±10}{2} when ± is minus. Subtract 10 from -20.

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

Divide -30 by 2.

$$x=-15$$

The equation is now solved.

$$x=-5 x=-15$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+10=5 x+10=-5$$

Subtract 10 from both sides of the equation.

$$x=-5 x=-15$$