Solve x^2+(x+6)^2=468

$$x ^ { 2 } + ( x + 6 ) ^ { 2 } = 468$$

Solve for x

$$x=-18$$

$$x=12$$

Steps Using Factoring By Grouping

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

$$x^{2}+x^{2}+12x+36=468$$

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

$$2x^{2}+12x+36=468$$

Subtract 468 from both sides.

$$2x^{2}+12x+36-468=0$$

Subtract 468 from 36 to get -432.

$$2x^{2}+12x-432=0$$

Divide both sides by 2.

$$x^{2}+6x-216=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-216. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-216\right)=-216$$

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 -216.

$$-1,216 -2,108 -3,72 -4,54 -6,36 -8,27 -9,24 -12,18$$

Calculate the sum for each pair.

$$-1+216=215 -2+108=106 -3+72=69 -4+54=50 -6+36=30 -8+27=19 -9+24=15 -12+18=6$$

The solution is the pair that gives sum 6.

$$a=-12 b=18$$

Rewrite x^{2}+6x-216 as \left(x^{2}-12x\right)+\left(18x-216\right).

$$\left(x^{2}-12x\right)+\left(18x-216\right)$$

Factor out x in the first and 18 in the second group.

$$x\left(x-12\right)+18\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x+18\right)$$

To find equation solutions, solve x-12=0 and x+18=0.

$$x=12 x=-18$$

Steps Using the Quadratic Formula

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

$$x^{2}+x^{2}+12x+36=468$$

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

$$2x^{2}+12x+36=468$$

Subtract 468 from both sides.

$$2x^{2}+12x+36-468=0$$

Subtract 468 from 36 to get -432.

$$2x^{2}+12x-432=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 12 for b, and -432 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-432\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-432\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-432\right)}}{2\times 2}$$

Multiply -8 times -432.

$$x=\frac{-12±\sqrt{144+3456}}{2\times 2}$$

Add 144 to 3456.

$$x=\frac{-12±\sqrt{3600}}{2\times 2}$$

Take the square root of 3600.

$$x=\frac{-12±60}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±60}{4}$$

Now solve the equation x=\frac{-12±60}{4} when ± is plus. Add -12 to 60.

$$x=\frac{48}{4}$$

Divide 48 by 4.

$$x=12$$

Now solve the equation x=\frac{-12±60}{4} when ± is minus. Subtract 60 from -12.

$$x=-\frac{72}{4}$$

Divide -72 by 4.

$$x=-18$$

The equation is now solved.

$$x=12 x=-18$$

Steps for Completing the Square

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

$$x^{2}+x^{2}+12x+36=468$$

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

$$2x^{2}+12x+36=468$$

Subtract 36 from both sides.

$$2x^{2}+12x=468-36$$

Subtract 36 from 468 to get 432.

$$2x^{2}+12x=432$$

Divide both sides by 2.

$$\frac{2x^{2}+12x}{2}=\frac{432}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{12}{2}x=\frac{432}{2}$$

Divide 12 by 2.

$$x^{2}+6x=\frac{432}{2}$$

Divide 432 by 2.

$$x^{2}+6x=216$$

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.

$$x^{2}+6x+3^{2}=216+3^{2}$$

Square 3.

$$x^{2}+6x+9=216+9$$

Add 216 to 9.

$$x^{2}+6x+9=225$$

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}.

$$\left(x+3\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{225}$$

Simplify.

$$x+3=15 x+3=-15$$

Subtract 3 from both sides of the equation.

$$x=12 x=-18$$