Solve 2500=1600+left(6+x+xright)^2

$$2500=1600+ { \left(6+x+x \right) }^{ 2 }$$

Solve for x

$$x=12$$

$$x=-18$$

Steps Using Factoring By Grouping

Combine x and x to get 2x.

$$2500=1600+\left(6+2x\right)^{2}$$

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

$$2500=1600+36+24x+4x^{2}$$

Add 1600 and 36 to get 1636.

$$2500=1636+24x+4x^{2}$$

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

$$1636+24x+4x^{2}=2500$$

Subtract 2500 from both sides.

$$1636+24x+4x^{2}-2500=0$$

Subtract 2500 from 1636 to get -864.

$$-864+24x+4x^{2}=0$$

Divide both sides by 4.

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

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

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

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

Combine x and x to get 2x.

$$2500=1600+\left(6+2x\right)^{2}$$

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

$$2500=1600+36+24x+4x^{2}$$

Add 1600 and 36 to get 1636.

$$2500=1636+24x+4x^{2}$$

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

$$1636+24x+4x^{2}=2500$$

Subtract 2500 from both sides.

$$1636+24x+4x^{2}-2500=0$$

Subtract 2500 from 1636 to get -864.

$$-864+24x+4x^{2}=0$$

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.

$$4x^{2}+24x-864=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 24 for b, and -864 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 4\left(-864\right)}}{2\times 4}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 4\left(-864\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-24±\sqrt{576-16\left(-864\right)}}{2\times 4}$$

Multiply -16 times -864.

$$x=\frac{-24±\sqrt{576+13824}}{2\times 4}$$

Add 576 to 13824.

$$x=\frac{-24±\sqrt{14400}}{2\times 4}$$

Take the square root of 14400.

$$x=\frac{-24±120}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-24±120}{8}$$

Now solve the equation x=\frac{-24±120}{8} when ± is plus. Add -24 to 120.

$$x=\frac{96}{8}$$

Divide 96 by 8.

$$x=12$$

Now solve the equation x=\frac{-24±120}{8} when ± is minus. Subtract 120 from -24.

$$x=-\frac{144}{8}$$

Divide -144 by 8.

$$x=-18$$

The equation is now solved.

$$x=12 x=-18$$

Steps for Completing the Square

Combine x and x to get 2x.

$$2500=1600+\left(6+2x\right)^{2}$$

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

$$2500=1600+36+24x+4x^{2}$$

Add 1600 and 36 to get 1636.

$$2500=1636+24x+4x^{2}$$

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

$$1636+24x+4x^{2}=2500$$

Subtract 1636 from both sides.

$$24x+4x^{2}=2500-1636$$

Subtract 1636 from 2500 to get 864.

$$24x+4x^{2}=864$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}+24x=864$$

Divide both sides by 4.

$$\frac{4x^{2}+24x}{4}=\frac{864}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{24}{4}x=\frac{864}{4}$$

Divide 24 by 4.

$$x^{2}+6x=\frac{864}{4}$$

Divide 864 by 4.

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

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.

$$x=12 x=-18$$