Solve (x+1)^2=256

$$( x + 1 ) ^ { 2 } = 256$$

Solve for x

$$x=15$$

$$x=-17$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}+2x+1=256$$

Subtract 256 from both sides.

$$x^{2}+2x+1-256=0$$

Subtract 256 from 1 to get -255.

$$x^{2}+2x-255=0$$

To solve the equation, factor x^{2}+2x-255 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.

$$a+b=2 ab=-255$$

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 -255.

$$-1,255 -3,85 -5,51 -15,17$$

Calculate the sum for each pair.

$$-1+255=254 -3+85=82 -5+51=46 -15+17=2$$

The solution is the pair that gives sum 2.

$$a=-15 b=17$$

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

$$\left(x-15\right)\left(x+17\right)$$

To find equation solutions, solve x-15=0 and x+17=0.

$$x=15 x=-17$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}+2x+1=256$$

Subtract 256 from both sides.

$$x^{2}+2x+1-256=0$$

Subtract 256 from 1 to get -255.

$$x^{2}+2x-255=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-255. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-255\right)=-255$$

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 -255.

$$-1,255 -3,85 -5,51 -15,17$$

Calculate the sum for each pair.

$$-1+255=254 -3+85=82 -5+51=46 -15+17=2$$

The solution is the pair that gives sum 2.

$$a=-15 b=17$$

Rewrite x^{2}+2x-255 as \left(x^{2}-15x\right)+\left(17x-255\right).

$$\left(x^{2}-15x\right)+\left(17x-255\right)$$

Factor out x in the first and 17 in the second group.

$$x\left(x-15\right)+17\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x+17\right)$$

To find equation solutions, solve x-15=0 and x+17=0.

$$x=15 x=-17$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}+2x+1=256$$

Subtract 256 from both sides.

$$x^{2}+2x+1-256=0$$

Subtract 256 from 1 to get -255.

$$x^{2}+2x-255=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -255 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-255\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-255\right)}}{2}$$

Multiply -4 times -255.

$$x=\frac{-2±\sqrt{4+1020}}{2}$$

Add 4 to 1020.

$$x=\frac{-2±\sqrt{1024}}{2}$$

Take the square root of 1024.

$$x=\frac{-2±32}{2}$$

Now solve the equation x=\frac{-2±32}{2} when ± is plus. Add -2 to 32.

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

Divide 30 by 2.

$$x=15$$

Now solve the equation x=\frac{-2±32}{2} when ± is minus. Subtract 32 from -2.

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

Divide -34 by 2.

$$x=-17$$

The equation is now solved.

$$x=15 x=-17$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{256}$$

Simplify.

$$x+1=16 x+1=-16$$

Subtract 1 from both sides of the equation.

$$x=15 x=-17$$