Solve 5(x-1)^2=125

$$5 ( x - 1 ) ^ { 2 } = 125$$

Solve for x

$$x=6$$

$$x=-4$$

Steps Using Factoring

Divide both sides by 5.

$$\left(x-1\right)^{2}=\frac{125}{5}$$

Divide 125 by 5 to get 25.

$$\left(x-1\right)^{2}=25$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=25$$

Subtract 25 from both sides.

$$x^{2}-2x+1-25=0$$

Subtract 25 from 1 to get -24.

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

To solve the equation, factor x^{2}-2x-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.

$$a+b=-2 ab=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

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

$$\left(x-6\right)\left(x+4\right)$$

To find equation solutions, solve x-6=0 and x+4=0.

$$x=6 x=-4$$

Steps Using Factoring By Grouping

Divide both sides by 5.

$$\left(x-1\right)^{2}=\frac{125}{5}$$

Divide 125 by 5 to get 25.

$$\left(x-1\right)^{2}=25$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=25$$

Subtract 25 from both sides.

$$x^{2}-2x+1-25=0$$

Subtract 25 from 1 to get -24.

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

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.

$$a+b=-2 ab=1\left(-24\right)=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

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

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite x^{2}-2x-24 as \left(x^{2}-6x\right)+\left(4x-24\right).

$$\left(x^{2}-6x\right)+\left(4x-24\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-6\right)+4\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+4\right)$$

To find equation solutions, solve x-6=0 and x+4=0.

$$x=6 x=-4$$

Steps Using the Quadratic Formula

Divide both sides by 5.

$$\left(x-1\right)^{2}=\frac{125}{5}$$

Divide 125 by 5 to get 25.

$$\left(x-1\right)^{2}=25$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=25$$

Subtract 25 from both sides.

$$x^{2}-2x+1-25=0$$

Subtract 25 from 1 to get -24.

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

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

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

Square -2.

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

Multiply -4 times -24.

$$x=\frac{-\left(-2\right)±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$x=\frac{-\left(-2\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-2\right)±10}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±10}{2}$$

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

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

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

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=6 x=-4$$

Steps for Completing the Square

Divide both sides by 5.

$$\left(x-1\right)^{2}=\frac{125}{5}$$

Divide 125 by 5 to get 25.

$$\left(x-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-1=5 x-1=-5$$

Add 1 to both sides of the equation.

$$x=6 x=-4$$