Solve x^2+(x+1)^2=365

$$x ^ { 2 } + ( x + 1 ) ^ { 2 } = 365$$

Solve for x

$$x=-14$$

$$x=13$$

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}.

$$x^{2}+x^{2}+2x+1=365$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+2x+1=365$$

Subtract 365 from both sides.

$$2x^{2}+2x+1-365=0$$

Subtract 365 from 1 to get -364.

$$2x^{2}+2x-364=0$$

Divide both sides by 2.

$$x^{2}+x-182=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-182. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-182\right)=-182$$

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 -182.

$$-1,182 -2,91 -7,26 -13,14$$

Calculate the sum for each pair.

$$-1+182=181 -2+91=89 -7+26=19 -13+14=1$$

The solution is the pair that gives sum 1.

$$a=-13 b=14$$

Rewrite x^{2}+x-182 as \left(x^{2}-13x\right)+\left(14x-182\right).

$$\left(x^{2}-13x\right)+\left(14x-182\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x-13\right)+14\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x+14\right)$$

To find equation solutions, solve x-13=0 and x+14=0.

$$x=13 x=-14$$

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}.

$$x^{2}+x^{2}+2x+1=365$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+2x+1=365$$

Subtract 365 from both sides.

$$2x^{2}+2x+1-365=0$$

Subtract 365 from 1 to get -364.

$$2x^{2}+2x-364=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 2 for b, and -364 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 2\left(-364\right)}}{2\times 2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 2\left(-364\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-2±\sqrt{4-8\left(-364\right)}}{2\times 2}$$

Multiply -8 times -364.

$$x=\frac{-2±\sqrt{4+2912}}{2\times 2}$$

Add 4 to 2912.

$$x=\frac{-2±\sqrt{2916}}{2\times 2}$$

Take the square root of 2916.

$$x=\frac{-2±54}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±54}{4}$$

Now solve the equation x=\frac{-2±54}{4} when ± is plus. Add -2 to 54.

$$x=\frac{52}{4}$$

Divide 52 by 4.

$$x=13$$

Now solve the equation x=\frac{-2±54}{4} when ± is minus. Subtract 54 from -2.

$$x=-\frac{56}{4}$$

Divide -56 by 4.

$$x=-14$$

The equation is now solved.

$$x=13 x=-14$$

Steps for Completing the Square

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

$$x^{2}+x^{2}+2x+1=365$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+2x+1=365$$

Subtract 1 from both sides.

$$2x^{2}+2x=365-1$$

Subtract 1 from 365 to get 364.

$$2x^{2}+2x=364$$

Divide both sides by 2.

$$\frac{2x^{2}+2x}{2}=\frac{364}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{2}{2}x=\frac{364}{2}$$

Divide 2 by 2.

$$x^{2}+x=\frac{364}{2}$$

Divide 364 by 2.

$$x^{2}+x=182$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=182+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=182+\frac{1}{4}$$

Add 182 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{729}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{729}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{27}{2} x+\frac{1}{2}=-\frac{27}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=13 x=-14$$