Solve 750^2=x^2+110-2x/2

$${ 750 }^{ 2 } = { x }^{ 2 } + \frac{ 110-2x }{ 2 }$$

Solve for x

$$x = \frac{\sqrt{2249781} + 1}{2} \approx 750.463499112$$

$$x=\frac{1-\sqrt{2249781}}{2}\approx -749.463499112$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\times 750^{2}=2x^{2}+110-2x$$

Calculate 750 to the power of 2 and get 562500.

$$2\times 562500=2x^{2}+110-2x$$

Multiply 2 and 562500 to get 1125000.

$$1125000=2x^{2}+110-2x$$

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

$$2x^{2}+110-2x=1125000$$

Subtract 1125000 from both sides.

$$2x^{2}+110-2x-1125000=0$$

Subtract 1125000 from 110 to get -1124890.

$$2x^{2}-1124890-2x=0$$

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.

$$2x^{2}-2x-1124890=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and -1124890 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-1124890\right)}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-1124890\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-1124890\right)}}{2\times 2}$$

Multiply -8 times -1124890.

$$x=\frac{-\left(-2\right)±\sqrt{4+8999120}}{2\times 2}$$

Add 4 to 8999120.

$$x=\frac{-\left(-2\right)±\sqrt{8999124}}{2\times 2}$$

Take the square root of 8999124.

$$x=\frac{-\left(-2\right)±2\sqrt{2249781}}{2\times 2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{2249781}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±2\sqrt{2249781}}{4}$$

Now solve the equation x=\frac{2±2\sqrt{2249781}}{4} when ± is plus. Add 2 to 2\sqrt{2249781}.

$$x=\frac{2\sqrt{2249781}+2}{4}$$

Divide 2+2\sqrt{2249781} by 4.

$$x=\frac{\sqrt{2249781}+1}{2}$$

Now solve the equation x=\frac{2±2\sqrt{2249781}}{4} when ± is minus. Subtract 2\sqrt{2249781} from 2.

$$x=\frac{2-2\sqrt{2249781}}{4}$$

Divide 2-2\sqrt{2249781} by 4.

$$x=\frac{1-\sqrt{2249781}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2249781}+1}{2} x=\frac{1-\sqrt{2249781}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\times 750^{2}=2x^{2}+110-2x$$

Calculate 750 to the power of 2 and get 562500.

$$2\times 562500=2x^{2}+110-2x$$

Multiply 2 and 562500 to get 1125000.

$$1125000=2x^{2}+110-2x$$

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

$$2x^{2}+110-2x=1125000$$

Subtract 110 from both sides.

$$2x^{2}-2x=1125000-110$$

Subtract 110 from 1125000 to get 1124890.

$$2x^{2}-2x=1124890$$

Divide both sides by 2.

$$\frac{2x^{2}-2x}{2}=\frac{1124890}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{2}{2}\right)x=\frac{1124890}{2}$$

Divide -2 by 2.

$$x^{2}-x=\frac{1124890}{2}$$

Divide 1124890 by 2.

$$x^{2}-x=562445$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=562445+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=562445+\frac{1}{4}$$

Add 562445 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{2249781}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{2249781}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{2249781}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{2249781}}{2} x-\frac{1}{2}=-\frac{\sqrt{2249781}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{2249781}+1}{2} x=\frac{1-\sqrt{2249781}}{2}$$