Solve 750=x^2+{left(110-2x/2right)}^2

$$750= { x }^{ 2 } + { \left( \frac{ 110-2x }{ 2 } \right) }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{-5\sqrt{61}i+55}{2}\approx 27.5-19.52562419i$$

$$x=\frac{55+5\sqrt{61}i}{2}\approx 27.5+19.52562419i$$

Steps Using the Quadratic Formula

Divide each term of 110-2x by 2 to get 55-x.

$$750=x^{2}+\left(55-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(55-x\right)^{2}.

$$750=x^{2}+3025-110x+x^{2}$$

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

$$750=2x^{2}+3025-110x$$

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

$$2x^{2}+3025-110x=750$$

Subtract 750 from both sides.

$$2x^{2}+3025-110x-750=0$$

Subtract 750 from 3025 to get 2275.

$$2x^{2}+2275-110x=0$$

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.

$$2x^{2}-110x+2275=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -110 for b, and 2275 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\times 2\times 2275}}{2\times 2}$$

Square -110.

$$x=\frac{-\left(-110\right)±\sqrt{12100-4\times 2\times 2275}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-110\right)±\sqrt{12100-8\times 2275}}{2\times 2}$$

Multiply -8 times 2275.

$$x=\frac{-\left(-110\right)±\sqrt{12100-18200}}{2\times 2}$$

Add 12100 to -18200.

$$x=\frac{-\left(-110\right)±\sqrt{-6100}}{2\times 2}$$

Take the square root of -6100.

$$x=\frac{-\left(-110\right)±10\sqrt{61}i}{2\times 2}$$

The opposite of -110 is 110.

$$x=\frac{110±10\sqrt{61}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{110±10\sqrt{61}i}{4}$$

Now solve the equation x=\frac{110±10\sqrt{61}i}{4} when ± is plus. Add 110 to 10i\sqrt{61}.

$$x=\frac{110+10\sqrt{61}i}{4}$$

Divide 110+10i\sqrt{61} by 4.

$$x=\frac{55+5\sqrt{61}i}{2}$$

Now solve the equation x=\frac{110±10\sqrt{61}i}{4} when ± is minus. Subtract 10i\sqrt{61} from 110.

$$x=\frac{-10\sqrt{61}i+110}{4}$$

Divide 110-10i\sqrt{61} by 4.

$$x=\frac{-5\sqrt{61}i+55}{2}$$

The equation is now solved.

$$x=\frac{55+5\sqrt{61}i}{2} x=\frac{-5\sqrt{61}i+55}{2}$$

Steps for Completing the Square

Divide each term of 110-2x by 2 to get 55-x.

$$750=x^{2}+\left(55-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(55-x\right)^{2}.

$$750=x^{2}+3025-110x+x^{2}$$

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

$$750=2x^{2}+3025-110x$$

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

$$2x^{2}+3025-110x=750$$

Subtract 3025 from both sides.

$$2x^{2}-110x=750-3025$$

Subtract 3025 from 750 to get -2275.

$$2x^{2}-110x=-2275$$

Divide both sides by 2.

$$\frac{2x^{2}-110x}{2}=-\frac{2275}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{110}{2}\right)x=-\frac{2275}{2}$$

Divide -110 by 2.

$$x^{2}-55x=-\frac{2275}{2}$$

Divide -55, the coefficient of the x term, by 2 to get -\frac{55}{2}. Then add the square of -\frac{55}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-55x+\left(-\frac{55}{2}\right)^{2}=-\frac{2275}{2}+\left(-\frac{55}{2}\right)^{2}$$

Square -\frac{55}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-55x+\frac{3025}{4}=-\frac{2275}{2}+\frac{3025}{4}$$

Add -\frac{2275}{2} to \frac{3025}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-55x+\frac{3025}{4}=-\frac{1525}{4}$$

Factor x^{2}-55x+\frac{3025}{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}.

$$\left(x-\frac{55}{2}\right)^{2}=-\frac{1525}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{2}\right)^{2}}=\sqrt{-\frac{1525}{4}}$$

Simplify.

$$x-\frac{55}{2}=\frac{5\sqrt{61}i}{2} x-\frac{55}{2}=-\frac{5\sqrt{61}i}{2}$$

Add \frac{55}{2} to both sides of the equation.

$$x=\frac{55+5\sqrt{61}i}{2} x=\frac{-5\sqrt{61}i+55}{2}$$