Solve 515=x^2+left(x+2right)^2+left(x+4right)^2

$$515= { x }^{ 2 } + { \left(x+2 \right) }^{ 2 } + { \left(x+4 \right) }^{ 2 }$$

Solve for x

$$x=-15$$

$$x=11$$

Steps Using Factoring By Grouping

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

$$515=x^{2}+x^{2}+4x+4+\left(x+4\right)^{2}$$

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

$$515=2x^{2}+4x+4+\left(x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$515=2x^{2}+4x+4+x^{2}+8x+16$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$515=3x^{2}+4x+4+8x+16$$

Combine 4x and 8x to get 12x.

$$515=3x^{2}+12x+4+16$$

Add 4 and 16 to get 20.

$$515=3x^{2}+12x+20$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+12x+20=515$$

Subtract 515 from both sides.

$$3x^{2}+12x+20-515=0$$

Subtract 515 from 20 to get -495.

$$3x^{2}+12x-495=0$$

Divide both sides by 3.

$$x^{2}+4x-165=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-165. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-165\right)=-165$$

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 -165.

$$-1,165 -3,55 -5,33 -11,15$$

Calculate the sum for each pair.

$$-1+165=164 -3+55=52 -5+33=28 -11+15=4$$

The solution is the pair that gives sum 4.

$$a=-11 b=15$$

Rewrite x^{2}+4x-165 as \left(x^{2}-11x\right)+\left(15x-165\right).

$$\left(x^{2}-11x\right)+\left(15x-165\right)$$

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

$$x\left(x-11\right)+15\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x+15\right)$$

To find equation solutions, solve x-11=0 and x+15=0.

$$x=11 x=-15$$

Steps Using the Quadratic Formula

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

$$515=x^{2}+x^{2}+4x+4+\left(x+4\right)^{2}$$

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

$$515=2x^{2}+4x+4+\left(x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$515=2x^{2}+4x+4+x^{2}+8x+16$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$515=3x^{2}+4x+4+8x+16$$

Combine 4x and 8x to get 12x.

$$515=3x^{2}+12x+4+16$$

Add 4 and 16 to get 20.

$$515=3x^{2}+12x+20$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+12x+20=515$$

Subtract 515 from both sides.

$$3x^{2}+12x+20-515=0$$

Subtract 515 from 20 to get -495.

$$3x^{2}+12x-495=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 12 for b, and -495 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 3\left(-495\right)}}{2\times 3}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 3\left(-495\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-12±\sqrt{144-12\left(-495\right)}}{2\times 3}$$

Multiply -12 times -495.

$$x=\frac{-12±\sqrt{144+5940}}{2\times 3}$$

Add 144 to 5940.

$$x=\frac{-12±\sqrt{6084}}{2\times 3}$$

Take the square root of 6084.

$$x=\frac{-12±78}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-12±78}{6}$$

Now solve the equation x=\frac{-12±78}{6} when ± is plus. Add -12 to 78.

$$x=\frac{66}{6}$$

Divide 66 by 6.

$$x=11$$

Now solve the equation x=\frac{-12±78}{6} when ± is minus. Subtract 78 from -12.

$$x=-\frac{90}{6}$$

Divide -90 by 6.

$$x=-15$$

The equation is now solved.

$$x=11 x=-15$$

Steps for Completing the Square

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

$$515=x^{2}+x^{2}+4x+4+\left(x+4\right)^{2}$$

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

$$515=2x^{2}+4x+4+\left(x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$515=2x^{2}+4x+4+x^{2}+8x+16$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$515=3x^{2}+4x+4+8x+16$$

Combine 4x and 8x to get 12x.

$$515=3x^{2}+12x+4+16$$

Add 4 and 16 to get 20.

$$515=3x^{2}+12x+20$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+12x+20=515$$

Subtract 20 from both sides.

$$3x^{2}+12x=515-20$$

Subtract 20 from 515 to get 495.

$$3x^{2}+12x=495$$

Divide both sides by 3.

$$\frac{3x^{2}+12x}{3}=\frac{495}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{12}{3}x=\frac{495}{3}$$

Divide 12 by 3.

$$x^{2}+4x=\frac{495}{3}$$

Divide 495 by 3.

$$x^{2}+4x=165$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=165+2^{2}$$

Square 2.

$$x^{2}+4x+4=165+4$$

Add 165 to 4.

$$x^{2}+4x+4=169$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{169}$$

Simplify.

$$x+2=13 x+2=-13$$

Subtract 2 from both sides of the equation.

$$x=11 x=-15$$