Solve (x-7)^2=100

$$( x - 7 ) ^ { 2 } = 100$$

Solve for x

$$x=17$$

$$x=-3$$

Steps Using Factoring

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

$$x^{2}-14x+49=100$$

Subtract 100 from both sides.

$$x^{2}-14x+49-100=0$$

Subtract 100 from 49 to get -51.

$$x^{2}-14x-51=0$$

To solve the equation, factor x^{2}-14x-51 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.

$$a+b=-14 ab=-51$$

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 -51.

$$1,-51 3,-17$$

Calculate the sum for each pair.

$$1-51=-50 3-17=-14$$

The solution is the pair that gives sum -14.

$$a=-17 b=3$$

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

$$\left(x-17\right)\left(x+3\right)$$

To find equation solutions, solve x-17=0 and x+3=0.

$$x=17 x=-3$$

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}.

$$x^{2}-14x+49=100$$

Subtract 100 from both sides.

$$x^{2}-14x+49-100=0$$

Subtract 100 from 49 to get -51.

$$x^{2}-14x-51=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-51. To find a and b, set up a system to be solved.

$$a+b=-14 ab=1\left(-51\right)=-51$$

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 -51.

$$1,-51 3,-17$$

Calculate the sum for each pair.

$$1-51=-50 3-17=-14$$

The solution is the pair that gives sum -14.

$$a=-17 b=3$$

Rewrite x^{2}-14x-51 as \left(x^{2}-17x\right)+\left(3x-51\right).

$$\left(x^{2}-17x\right)+\left(3x-51\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(x-17\right)+3\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(x+3\right)$$

To find equation solutions, solve x-17=0 and x+3=0.

$$x=17 x=-3$$

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}.

$$x^{2}-14x+49=100$$

Subtract 100 from both sides.

$$x^{2}-14x+49-100=0$$

Subtract 100 from 49 to get -51.

$$x^{2}-14x-51=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and -51 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-51\right)}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-51\right)}}{2}$$

Multiply -4 times -51.

$$x=\frac{-\left(-14\right)±\sqrt{196+204}}{2}$$

Add 196 to 204.

$$x=\frac{-\left(-14\right)±\sqrt{400}}{2}$$

Take the square root of 400.

$$x=\frac{-\left(-14\right)±20}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±20}{2}$$

Now solve the equation x=\frac{14±20}{2} when ± is plus. Add 14 to 20.

$$x=\frac{34}{2}$$

Divide 34 by 2.

$$x=17$$

Now solve the equation x=\frac{14±20}{2} when ± is minus. Subtract 20 from 14.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=17 x=-3$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-7=10 x-7=-10$$

Add 7 to both sides of the equation.

$$x=17 x=-3$$