Solve (x-7)^2=16

$$( x - 7 ) ^ { 2 } = 16$$

Solve for x

$$x=11$$

$$x=3$$

Steps Using Factoring

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

$$x^{2}-14x+49=16$$

Subtract 16 from both sides.

$$x^{2}-14x+49-16=0$$

Subtract 16 from 49 to get 33.

$$x^{2}-14x+33=0$$

To solve the equation, factor x^{2}-14x+33 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.

$$a+b=-14 ab=33$$

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 33.

$$-1,-33 -3,-11$$

Calculate the sum for each pair.

$$-1-33=-34 -3-11=-14$$

The solution is the pair that gives sum -14.

$$a=-11 b=-3$$

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

$$\left(x-11\right)\left(x-3\right)$$

To find equation solutions, solve x-11=0 and x-3=0.

$$x=11 x=3$$

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}.

$$x^{2}-14x+49=16$$

Subtract 16 from both sides.

$$x^{2}-14x+49-16=0$$

Subtract 16 from 49 to get 33.

$$x^{2}-14x+33=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+33. To find a and b, set up a system to be solved.

$$a+b=-14 ab=1\times 33=33$$

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 33.

$$-1,-33 -3,-11$$

Calculate the sum for each pair.

$$-1-33=-34 -3-11=-14$$

The solution is the pair that gives sum -14.

$$a=-11 b=-3$$

Rewrite x^{2}-14x+33 as \left(x^{2}-11x\right)+\left(-3x+33\right).

$$\left(x^{2}-11x\right)+\left(-3x+33\right)$$

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

$$x\left(x-11\right)-3\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x-3\right)$$

To find equation solutions, solve x-11=0 and x-3=0.

$$x=11 x=3$$

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}.

$$x^{2}-14x+49=16$$

Subtract 16 from both sides.

$$x^{2}-14x+49-16=0$$

Subtract 16 from 49 to get 33.

$$x^{2}-14x+33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 33}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 33}}{2}$$

Multiply -4 times 33.

$$x=\frac{-\left(-14\right)±\sqrt{196-132}}{2}$$

Add 196 to -132.

$$x=\frac{-\left(-14\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-\left(-14\right)±8}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±8}{2}$$

Now solve the equation x=\frac{14±8}{2} when ± is plus. Add 14 to 8.

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

Divide 22 by 2.

$$x=11$$

Now solve the equation x=\frac{14±8}{2} when ± is minus. Subtract 8 from 14.

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

Divide 6 by 2.

$$x=3$$

The equation is now solved.

$$x=11 x=3$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-7=4 x-7=-4$$

Add 7 to both sides of the equation.

$$x=11 x=3$$