Solve (x+7)^2-64=0

$$( x + 7 ) ^ { 2 } - 64 = 0$$

Solve for x

$$x=1$$

$$x=-15$$

Steps Using Factoring

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

$$x^{2}+14x+49-64=0$$

Subtract 64 from 49 to get -15.

$$x^{2}+14x-15=0$$

To solve the equation, factor x^{2}+14x-15 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.

$$a+b=14 ab=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 14.

$$a=-1 b=15$$

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

$$\left(x-1\right)\left(x+15\right)$$

To find equation solutions, solve x-1=0 and x+15=0.

$$x=1 x=-15$$

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}.

$$x^{2}+14x+49-64=0$$

Subtract 64 from 49 to get -15.

$$x^{2}+14x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-15\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 14.

$$a=-1 b=15$$

Rewrite x^{2}+14x-15 as \left(x^{2}-x\right)+\left(15x-15\right).

$$\left(x^{2}-x\right)+\left(15x-15\right)$$

Factor out x in the first and 15 in the second group.

$$x\left(x-1\right)+15\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+15\right)$$

To find equation solutions, solve x-1=0 and x+15=0.

$$x=1 x=-15$$

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}.

$$x^{2}+14x+49-64=0$$

Subtract 64 from 49 to get -15.

$$x^{2}+14x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-15\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{-14±\sqrt{196+60}}{2}$$

Add 196 to 60.

$$x=\frac{-14±\sqrt{256}}{2}$$

Take the square root of 256.

$$x=\frac{-14±16}{2}$$

Now solve the equation x=\frac{-14±16}{2} when ± is plus. Add -14 to 16.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-14±16}{2} when ± is minus. Subtract 16 from -14.

$$x=-\frac{30}{2}$$

Divide -30 by 2.

$$x=-15$$

The equation is now solved.

$$x=1 x=-15$$

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}.

$$x^{2}+14x+49-64=0$$

Subtract 64 from 49 to get -15.

$$x^{2}+14x-15=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$x^{2}+14x=15$$

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.

$$x^{2}+14x+7^{2}=15+7^{2}$$

Square 7.

$$x^{2}+14x+49=15+49$$

Add 15 to 49.

$$x^{2}+14x+49=64$$

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}.

$$\left(x+7\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{64}$$

Simplify.

$$x+7=8 x+7=-8$$

Subtract 7 from both sides of the equation.

$$x=1 x=-15$$