Solve (x-5)^2-6(x-5)+5=0

$$( x - 5 ) ^ { 2 } - 6 ( x - 5 ) + 5 = 0$$

Solve for x

$$x=6$$

$$x=10$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}-10x+25-6\left(x-5\right)+5=0$$

Use the distributive property to multiply -6 by x-5.

$$x^{2}-10x+25-6x+30+5=0$$

Combine -10x and -6x to get -16x.

$$x^{2}-16x+25+30+5=0$$

Add 25 and 30 to get 55.

$$x^{2}-16x+55+5=0$$

Add 55 and 5 to get 60.

$$x^{2}-16x+60=0$$

To solve the equation, factor x^{2}-16x+60 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.

$$a+b=-16 ab=60$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 60.

$$-1,-60 -2,-30 -3,-20 -4,-15 -5,-12 -6,-10$$

Calculate the sum for each pair.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

The solution is the pair that gives sum -16.

$$a=-10 b=-6$$

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

$$\left(x-10\right)\left(x-6\right)$$

To find equation solutions, solve x-10=0 and x-6=0.

$$x=10 x=6$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}-10x+25-6\left(x-5\right)+5=0$$

Use the distributive property to multiply -6 by x-5.

$$x^{2}-10x+25-6x+30+5=0$$

Combine -10x and -6x to get -16x.

$$x^{2}-16x+25+30+5=0$$

Add 25 and 30 to get 55.

$$x^{2}-16x+55+5=0$$

Add 55 and 5 to get 60.

$$x^{2}-16x+60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+60. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\times 60=60$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 60.

$$-1,-60 -2,-30 -3,-20 -4,-15 -5,-12 -6,-10$$

Calculate the sum for each pair.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

The solution is the pair that gives sum -16.

$$a=-10 b=-6$$

Rewrite x^{2}-16x+60 as \left(x^{2}-10x\right)+\left(-6x+60\right).

$$\left(x^{2}-10x\right)+\left(-6x+60\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-10\right)-6\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x-6\right)$$

To find equation solutions, solve x-10=0 and x-6=0.

$$x=10 x=6$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}-10x+25-6\left(x-5\right)+5=0$$

Use the distributive property to multiply -6 by x-5.

$$x^{2}-10x+25-6x+30+5=0$$

Combine -10x and -6x to get -16x.

$$x^{2}-16x+25+30+5=0$$

Add 25 and 30 to get 55.

$$x^{2}-16x+55+5=0$$

Add 55 and 5 to get 60.

$$x^{2}-16x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 60}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 60}}{2}$$

Multiply -4 times 60.

$$x=\frac{-\left(-16\right)±\sqrt{256-240}}{2}$$

Add 256 to -240.

$$x=\frac{-\left(-16\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-16\right)±4}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±4}{2}$$

Now solve the equation x=\frac{16±4}{2} when ± is plus. Add 16 to 4.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{16±4}{2} when ± is minus. Subtract 4 from 16.

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

Divide 12 by 2.

$$x=6$$

The equation is now solved.

$$x=10 x=6$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-5\right)^{2}.

$$x^{2}-10x+25-6\left(x-5\right)+5=0$$

Use the distributive property to multiply -6 by x-5.

$$x^{2}-10x+25-6x+30+5=0$$

Combine -10x and -6x to get -16x.

$$x^{2}-16x+25+30+5=0$$

Add 25 and 30 to get 55.

$$x^{2}-16x+55+5=0$$

Add 55 and 5 to get 60.

$$x^{2}-16x+60=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-16x=-60$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-60+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-60+64$$

Add -60 to 64.

$$x^{2}-16x+64=4$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-8=2 x-8=-2$$

Add 8 to both sides of the equation.

$$x=10 x=6$$