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

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

Solve for x

$$x = \frac{3 \sqrt{249} + 63}{4} \approx 27.584800379$$

$$x = \frac{63 - 3 \sqrt{249}}{4} \approx 3.915199621$$

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.

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

Add 8x to both sides.

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

Combine 55x and 8x to get 63x.

$$63x-200-2x^{2}=16$$

Subtract 16 from both sides.

$$63x-200-2x^{2}-16=0$$

Subtract 16 from -200 to get -216.

$$63x-216-2x^{2}=0$$

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.

$$-2x^{2}+63x-216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 63 for b, and -216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-63±\sqrt{63^{2}-4\left(-2\right)\left(-216\right)}}{2\left(-2\right)}$$

Square 63.

$$x=\frac{-63±\sqrt{3969-4\left(-2\right)\left(-216\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-63±\sqrt{3969+8\left(-216\right)}}{2\left(-2\right)}$$

Multiply 8 times -216.

$$x=\frac{-63±\sqrt{3969-1728}}{2\left(-2\right)}$$

Add 3969 to -1728.

$$x=\frac{-63±\sqrt{2241}}{2\left(-2\right)}$$

Take the square root of 2241.

$$x=\frac{-63±3\sqrt{249}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-63±3\sqrt{249}}{-4}$$

Now solve the equation x=\frac{-63±3\sqrt{249}}{-4} when ± is plus. Add -63 to 3\sqrt{249}.

$$x=\frac{3\sqrt{249}-63}{-4}$$

Divide -63+3\sqrt{249} by -4.

$$x=\frac{63-3\sqrt{249}}{4}$$

Now solve the equation x=\frac{-63±3\sqrt{249}}{-4} when ± is minus. Subtract 3\sqrt{249} from -63.

$$x=\frac{-3\sqrt{249}-63}{-4}$$

Divide -63-3\sqrt{249} by -4.

$$x=\frac{3\sqrt{249}+63}{4}$$

The equation is now solved.

$$x=\frac{63-3\sqrt{249}}{4} x=\frac{3\sqrt{249}+63}{4}$$

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.

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

Add 8x to both sides.

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

Combine 55x and 8x to get 63x.

$$63x-200-2x^{2}=16$$

Add 200 to both sides.

$$63x-2x^{2}=16+200$$

Add 16 and 200 to get 216.

$$63x-2x^{2}=216$$

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.

$$-2x^{2}+63x=216$$

Divide both sides by -2.

$$\frac{-2x^{2}+63x}{-2}=\frac{216}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{63}{-2}x=\frac{216}{-2}$$

Divide 63 by -2.

$$x^{2}-\frac{63}{2}x=\frac{216}{-2}$$

Divide 216 by -2.

$$x^{2}-\frac{63}{2}x=-108$$

Divide -\frac{63}{2}, the coefficient of the x term, by 2 to get -\frac{63}{4}. Then add the square of -\frac{63}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{63}{2}x+\left(-\frac{63}{4}\right)^{2}=-108+\left(-\frac{63}{4}\right)^{2}$$

Square -\frac{63}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{63}{2}x+\frac{3969}{16}=-108+\frac{3969}{16}$$

Add -108 to \frac{3969}{16}.

$$x^{2}-\frac{63}{2}x+\frac{3969}{16}=\frac{2241}{16}$$

Factor x^{2}-\frac{63}{2}x+\frac{3969}{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}.

$$\left(x-\frac{63}{4}\right)^{2}=\frac{2241}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{63}{4}\right)^{2}}=\sqrt{\frac{2241}{16}}$$

Simplify.

$$x-\frac{63}{4}=\frac{3\sqrt{249}}{4} x-\frac{63}{4}=-\frac{3\sqrt{249}}{4}$$

Add \frac{63}{4} to both sides of the equation.

$$x=\frac{3\sqrt{249}+63}{4} x=\frac{63-3\sqrt{249}}{4}$$