Solve 55x+200=x^2+left(x+4right)^2

$$55x+200= { x }^{ 2 } + { \left(x+4 \right) }^{ 2 }$$

Solve for x

$$x = \frac{3 \sqrt{409} + 47}{4} \approx 26.917811312$$

$$x=\frac{47-3\sqrt{409}}{4}\approx -3.417811312$$

Steps Using the Quadratic Formula

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

$$55x+200=x^{2}+x^{2}+8x+16$$

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

$$55x+200=2x^{2}+8x+16$$

Subtract 2x^{2} from both sides.

$$55x+200-2x^{2}=8x+16$$

Subtract 8x from both sides.

$$55x+200-2x^{2}-8x=16$$

Combine 55x and -8x to get 47x.

$$47x+200-2x^{2}=16$$

Subtract 16 from both sides.

$$47x+200-2x^{2}-16=0$$

Subtract 16 from 200 to get 184.

$$47x+184-2x^{2}=0$$

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.

$$-2x^{2}+47x+184=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 47 for b, and 184 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-47±\sqrt{47^{2}-4\left(-2\right)\times 184}}{2\left(-2\right)}$$

Square 47.

$$x=\frac{-47±\sqrt{2209-4\left(-2\right)\times 184}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-47±\sqrt{2209+8\times 184}}{2\left(-2\right)}$$

Multiply 8 times 184.

$$x=\frac{-47±\sqrt{2209+1472}}{2\left(-2\right)}$$

Add 2209 to 1472.

$$x=\frac{-47±\sqrt{3681}}{2\left(-2\right)}$$

Take the square root of 3681.

$$x=\frac{-47±3\sqrt{409}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-47±3\sqrt{409}}{-4}$$

Now solve the equation x=\frac{-47±3\sqrt{409}}{-4} when ± is plus. Add -47 to 3\sqrt{409}.

$$x=\frac{3\sqrt{409}-47}{-4}$$

Divide -47+3\sqrt{409} by -4.

$$x=\frac{47-3\sqrt{409}}{4}$$

Now solve the equation x=\frac{-47±3\sqrt{409}}{-4} when ± is minus. Subtract 3\sqrt{409} from -47.

$$x=\frac{-3\sqrt{409}-47}{-4}$$

Divide -47-3\sqrt{409} by -4.

$$x=\frac{3\sqrt{409}+47}{4}$$

The equation is now solved.

$$x=\frac{47-3\sqrt{409}}{4} x=\frac{3\sqrt{409}+47}{4}$$

Steps for Completing the Square

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

$$55x+200=x^{2}+x^{2}+8x+16$$

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

$$55x+200=2x^{2}+8x+16$$

Subtract 2x^{2} from both sides.

$$55x+200-2x^{2}=8x+16$$

Subtract 8x from both sides.

$$55x+200-2x^{2}-8x=16$$

Combine 55x and -8x to get 47x.

$$47x+200-2x^{2}=16$$

Subtract 200 from both sides.

$$47x-2x^{2}=16-200$$

Subtract 200 from 16 to get -184.

$$47x-2x^{2}=-184$$

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.

$$-2x^{2}+47x=-184$$

Divide both sides by -2.

$$\frac{-2x^{2}+47x}{-2}=-\frac{184}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{47}{-2}x=-\frac{184}{-2}$$

Divide 47 by -2.

$$x^{2}-\frac{47}{2}x=-\frac{184}{-2}$$

Divide -184 by -2.

$$x^{2}-\frac{47}{2}x=92$$

Divide -\frac{47}{2}, the coefficient of the x term, by 2 to get -\frac{47}{4}. Then add the square of -\frac{47}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{47}{2}x+\left(-\frac{47}{4}\right)^{2}=92+\left(-\frac{47}{4}\right)^{2}$$

Square -\frac{47}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{47}{2}x+\frac{2209}{16}=92+\frac{2209}{16}$$

Add 92 to \frac{2209}{16}.

$$x^{2}-\frac{47}{2}x+\frac{2209}{16}=\frac{3681}{16}$$

Factor x^{2}-\frac{47}{2}x+\frac{2209}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{47}{4}\right)^{2}=\frac{3681}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{47}{4}\right)^{2}}=\sqrt{\frac{3681}{16}}$$

Simplify.

$$x-\frac{47}{4}=\frac{3\sqrt{409}}{4} x-\frac{47}{4}=-\frac{3\sqrt{409}}{4}$$

Add \frac{47}{4} to both sides of the equation.

$$x=\frac{3\sqrt{409}+47}{4} x=\frac{47-3\sqrt{409}}{4}$$