Solve x^2+(6+x)^2=68

$$x ^ { 2 } + ( 6 + x ) ^ { 2 } = 68$$

Solve for x

$$x=-8$$

$$x=2$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+x\right)^{2}.

$$x^{2}+36+12x+x^{2}=68$$

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

$$2x^{2}+36+12x=68$$

Subtract 68 from both sides.

$$2x^{2}+36+12x-68=0$$

Subtract 68 from 36 to get -32.

$$2x^{2}-32+12x=0$$

Divide both sides by 2.

$$x^{2}-16+6x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+6x-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-16\right)=-16$$

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 -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 6.

$$a=-2 b=8$$

Rewrite x^{2}+6x-16 as \left(x^{2}-2x\right)+\left(8x-16\right).

$$\left(x^{2}-2x\right)+\left(8x-16\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x-2\right)+8\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+8\right)$$

To find equation solutions, solve x-2=0 and x+8=0.

$$x=2 x=-8$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+x\right)^{2}.

$$x^{2}+36+12x+x^{2}=68$$

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

$$2x^{2}+36+12x=68$$

Subtract 68 from both sides.

$$2x^{2}+36+12x-68=0$$

Subtract 68 from 36 to get -32.

$$2x^{2}-32+12x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}+12x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 12 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-32\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-32\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-32\right)}}{2\times 2}$$

Multiply -8 times -32.

$$x=\frac{-12±\sqrt{144+256}}{2\times 2}$$

Add 144 to 256.

$$x=\frac{-12±\sqrt{400}}{2\times 2}$$

Take the square root of 400.

$$x=\frac{-12±20}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±20}{4}$$

Now solve the equation x=\frac{-12±20}{4} when ± is plus. Add -12 to 20.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-12±20}{4} when ± is minus. Subtract 20 from -12.

$$x=-\frac{32}{4}$$

Divide -32 by 4.

$$x=-8$$

The equation is now solved.

$$x=2 x=-8$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+x\right)^{2}.

$$x^{2}+36+12x+x^{2}=68$$

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

$$2x^{2}+36+12x=68$$

Subtract 36 from both sides.

$$2x^{2}+12x=68-36$$

Subtract 36 from 68 to get 32.

$$2x^{2}+12x=32$$

Divide both sides by 2.

$$\frac{2x^{2}+12x}{2}=\frac{32}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{12}{2}x=\frac{32}{2}$$

Divide 12 by 2.

$$x^{2}+6x=\frac{32}{2}$$

Divide 32 by 2.

$$x^{2}+6x=16$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=16+3^{2}$$

Square 3.

$$x^{2}+6x+9=16+9$$

Add 16 to 9.

$$x^{2}+6x+9=25$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+3=5 x+3=-5$$

Subtract 3 from both sides of the equation.

$$x=2 x=-8$$