Solve left(x-8right)^2-13left(x-8right)+30=0

$${ \left(x-8 \right) }^{ 2 } -13 \left( x-8 \right) +30=0$$

Solve for x

$$x=11$$

$$x=18$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64-13\left(x-8\right)+30=0$$

Use the distributive property to multiply -13 by x-8.

$$x^{2}-16x+64-13x+104+30=0$$

Combine -16x and -13x to get -29x.

$$x^{2}-29x+64+104+30=0$$

Add 64 and 104 to get 168.

$$x^{2}-29x+168+30=0$$

Add 168 and 30 to get 198.

$$x^{2}-29x+198=0$$

To solve the equation, factor x^{2}-29x+198 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.

$$a+b=-29 ab=198$$

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 198.

$$-1,-198 -2,-99 -3,-66 -6,-33 -9,-22 -11,-18$$

Calculate the sum for each pair.

$$-1-198=-199 -2-99=-101 -3-66=-69 -6-33=-39 -9-22=-31 -11-18=-29$$

The solution is the pair that gives sum -29.

$$a=-18 b=-11$$

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

$$\left(x-18\right)\left(x-11\right)$$

To find equation solutions, solve x-18=0 and x-11=0.

$$x=18 x=11$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64-13\left(x-8\right)+30=0$$

Use the distributive property to multiply -13 by x-8.

$$x^{2}-16x+64-13x+104+30=0$$

Combine -16x and -13x to get -29x.

$$x^{2}-29x+64+104+30=0$$

Add 64 and 104 to get 168.

$$x^{2}-29x+168+30=0$$

Add 168 and 30 to get 198.

$$x^{2}-29x+198=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+198. To find a and b, set up a system to be solved.

$$a+b=-29 ab=1\times 198=198$$

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 198.

$$-1,-198 -2,-99 -3,-66 -6,-33 -9,-22 -11,-18$$

Calculate the sum for each pair.

$$-1-198=-199 -2-99=-101 -3-66=-69 -6-33=-39 -9-22=-31 -11-18=-29$$

The solution is the pair that gives sum -29.

$$a=-18 b=-11$$

Rewrite x^{2}-29x+198 as \left(x^{2}-18x\right)+\left(-11x+198\right).

$$\left(x^{2}-18x\right)+\left(-11x+198\right)$$

Factor out x in the first and -11 in the second group.

$$x\left(x-18\right)-11\left(x-18\right)$$

Factor out common term x-18 by using distributive property.

$$\left(x-18\right)\left(x-11\right)$$

To find equation solutions, solve x-18=0 and x-11=0.

$$x=18 x=11$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64-13\left(x-8\right)+30=0$$

Use the distributive property to multiply -13 by x-8.

$$x^{2}-16x+64-13x+104+30=0$$

Combine -16x and -13x to get -29x.

$$x^{2}-29x+64+104+30=0$$

Add 64 and 104 to get 168.

$$x^{2}-29x+168+30=0$$

Add 168 and 30 to get 198.

$$x^{2}-29x+198=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -29 for b, and 198 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 198}}{2}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 198}}{2}$$

Multiply -4 times 198.

$$x=\frac{-\left(-29\right)±\sqrt{841-792}}{2}$$

Add 841 to -792.

$$x=\frac{-\left(-29\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$x=\frac{-\left(-29\right)±7}{2}$$

The opposite of -29 is 29.

$$x=\frac{29±7}{2}$$

Now solve the equation x=\frac{29±7}{2} when ± is plus. Add 29 to 7.

$$x=\frac{36}{2}$$

Divide 36 by 2.

$$x=18$$

Now solve the equation x=\frac{29±7}{2} when ± is minus. Subtract 7 from 29.

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

The equation is now solved.

$$x=18 x=11$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64-13\left(x-8\right)+30=0$$

Use the distributive property to multiply -13 by x-8.

$$x^{2}-16x+64-13x+104+30=0$$

Combine -16x and -13x to get -29x.

$$x^{2}-29x+64+104+30=0$$

Add 64 and 104 to get 168.

$$x^{2}-29x+168+30=0$$

Add 168 and 30 to get 198.

$$x^{2}-29x+198=0$$

Subtract 198 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-29x=-198$$

Divide -29, the coefficient of the x term, by 2 to get -\frac{29}{2}. Then add the square of -\frac{29}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-29x+\left(-\frac{29}{2}\right)^{2}=-198+\left(-\frac{29}{2}\right)^{2}$$

Square -\frac{29}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-29x+\frac{841}{4}=-198+\frac{841}{4}$$

Add -198 to \frac{841}{4}.

$$x^{2}-29x+\frac{841}{4}=\frac{49}{4}$$

Factor x^{2}-29x+\frac{841}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{29}{2}=\frac{7}{2} x-\frac{29}{2}=-\frac{7}{2}$$

Add \frac{29}{2} to both sides of the equation.

$$x=18 x=11$$