Solve (x-7)^2-8=17

$$( x - 7 ) ^ { 2 } - 8 = 17$$

Solve for x

$$x=12$$

$$x=2$$

Steps Using Factoring

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

$$x^{2}-14x+49-8=17$$

Subtract 8 from 49 to get 41.

$$x^{2}-14x+41=17$$

Subtract 17 from both sides.

$$x^{2}-14x+41-17=0$$

Subtract 17 from 41 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 negative, a and b are both negative. 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.

$$a=-12 b=-2$$

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

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

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

$$x=12 x=2$$

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}.

$$x^{2}-14x+49-8=17$$

Subtract 8 from 49 to get 41.

$$x^{2}-14x+41=17$$

Subtract 17 from both sides.

$$x^{2}-14x+41-17=0$$

Subtract 17 from 41 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 negative, a and b are both negative. 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.

$$a=-12 b=-2$$

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

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

Factor out x in the first and -2 in the second group.

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

Factor out common term x-12 by using distributive property.

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

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

$$x=12 x=2$$

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}.

$$x^{2}-14x+49-8=17$$

Subtract 8 from 49 to get 41.

$$x^{2}-14x+41=17$$

Subtract 17 from both sides.

$$x^{2}-14x+41-17=0$$

Subtract 17 from 41 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}.

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

Square -14.

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

Multiply -4 times 24.

$$x=\frac{-\left(-14\right)±\sqrt{196-96}}{2}$$

Add 196 to -96.

$$x=\frac{-\left(-14\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-14\right)±10}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±10}{2}$$

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

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

Divide 24 by 2.

$$x=12$$

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

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

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=12 x=2$$

Steps for Completing the Square

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

$$x^{2}-14x+49-8=17$$

Subtract 8 from 49 to get 41.

$$x^{2}-14x+41=17$$

Subtract 41 from both sides.

$$x^{2}-14x=17-41$$

Subtract 41 from 17 to get -24.

$$x^{2}-14x=-24$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-24+49$$

Add -24 to 49.

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

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 7 to both sides of the equation.

$$x=12 x=2$$