Solve 4(x+7)^2=16

$$4 ( x + 7 ) ^ { 2 } = 16$$

Solve for x

$$x=-5$$

$$x=-9$$

Steps Using Factoring

Divide both sides by 4.

$$\left(x+7\right)^{2}=\frac{16}{4}$$

Divide 16 by 4 to get 4.

$$\left(x+7\right)^{2}=4$$

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

$$x^{2}+14x+49=4$$

Subtract 4 from both sides.

$$x^{2}+14x+49-4=0$$

Subtract 4 from 49 to get 45.

$$x^{2}+14x+45=0$$

To solve the equation, factor x^{2}+14x+45 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.

$$a+b=14 ab=45$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 45.

$$1,45 3,15 5,9$$

Calculate the sum for each pair.

$$1+45=46 3+15=18 5+9=14$$

The solution is the pair that gives sum 14.

$$a=5 b=9$$

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

$$\left(x+5\right)\left(x+9\right)$$

To find equation solutions, solve x+5=0 and x+9=0.

$$x=-5 x=-9$$

Steps Using Factoring By Grouping

Divide both sides by 4.

$$\left(x+7\right)^{2}=\frac{16}{4}$$

Divide 16 by 4 to get 4.

$$\left(x+7\right)^{2}=4$$

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

$$x^{2}+14x+49=4$$

Subtract 4 from both sides.

$$x^{2}+14x+49-4=0$$

Subtract 4 from 49 to get 45.

$$x^{2}+14x+45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+45. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 45=45$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 45.

$$1,45 3,15 5,9$$

Calculate the sum for each pair.

$$1+45=46 3+15=18 5+9=14$$

The solution is the pair that gives sum 14.

$$a=5 b=9$$

Rewrite x^{2}+14x+45 as \left(x^{2}+5x\right)+\left(9x+45\right).

$$\left(x^{2}+5x\right)+\left(9x+45\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x+5\right)+9\left(x+5\right)$$

Factor out common term x+5 by using distributive property.

$$\left(x+5\right)\left(x+9\right)$$

To find equation solutions, solve x+5=0 and x+9=0.

$$x=-5 x=-9$$

Steps Using the Quadratic Formula

Divide both sides by 4.

$$\left(x+7\right)^{2}=\frac{16}{4}$$

Divide 16 by 4 to get 4.

$$\left(x+7\right)^{2}=4$$

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

$$x^{2}+14x+49=4$$

Subtract 4 from both sides.

$$x^{2}+14x+49-4=0$$

Subtract 4 from 49 to get 45.

$$x^{2}+14x+45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 45}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 45}}{2}$$

Multiply -4 times 45.

$$x=\frac{-14±\sqrt{196-180}}{2}$$

Add 196 to -180.

$$x=\frac{-14±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-14±4}{2}$$

Now solve the equation x=\frac{-14±4}{2} when ± is plus. Add -14 to 4.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

Now solve the equation x=\frac{-14±4}{2} when ± is minus. Subtract 4 from -14.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=-5 x=-9$$

Steps for Completing the Square

Divide both sides by 4.

$$\left(x+7\right)^{2}=\frac{16}{4}$$

Divide 16 by 4 to get 4.

$$\left(x+7\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{4}$$

Simplify.

$$x+7=2 x+7=-2$$

Subtract 7 from both sides of the equation.

$$x=-5 x=-9$$