Solve 76+x*(112.6-x)=x*x

$$76+x \times (112.6-x)=x \times x$$

Solve for x

$$x = \frac{\sqrt{332169} + 563}{20} \approx 56.967052243$$

$$x=\frac{563-\sqrt{332169}}{20}\approx -0.667052243$$

Steps Using the Quadratic Formula

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

$$76+x\left(112.6-x\right)=x^{2}$$

Use the distributive property to multiply x by 112.6-x.

$$76+112.6x-x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$76+112.6x-x^{2}-x^{2}=0$$

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

$$76+112.6x-2x^{2}=0$$

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.

$$-2x^{2}+112.6x+76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 112.6 for b, and 76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112.6±\sqrt{112.6^{2}-4\left(-2\right)\times 76}}{2\left(-2\right)}$$

Square 112.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-112.6±\sqrt{12678.76-4\left(-2\right)\times 76}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-112.6±\sqrt{12678.76+8\times 76}}{2\left(-2\right)}$$

Multiply 8 times 76.

$$x=\frac{-112.6±\sqrt{12678.76+608}}{2\left(-2\right)}$$

Add 12678.76 to 608.

$$x=\frac{-112.6±\sqrt{13286.76}}{2\left(-2\right)}$$

Take the square root of 13286.76.

$$x=\frac{-112.6±\frac{\sqrt{332169}}{5}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-112.6±\frac{\sqrt{332169}}{5}}{-4}$$

Now solve the equation x=\frac{-112.6±\frac{\sqrt{332169}}{5}}{-4} when ± is plus. Add -112.6 to \frac{\sqrt{332169}}{5}.

$$x=\frac{\sqrt{332169}-563}{-4\times 5}$$

Divide \frac{-563+\sqrt{332169}}{5} by -4.

$$x=\frac{563-\sqrt{332169}}{20}$$

Now solve the equation x=\frac{-112.6±\frac{\sqrt{332169}}{5}}{-4} when ± is minus. Subtract \frac{\sqrt{332169}}{5} from -112.6.

$$x=\frac{-\sqrt{332169}-563}{-4\times 5}$$

Divide \frac{-563-\sqrt{332169}}{5} by -4.

$$x=\frac{\sqrt{332169}+563}{20}$$

The equation is now solved.

$$x=\frac{563-\sqrt{332169}}{20} x=\frac{\sqrt{332169}+563}{20}$$

Steps for Completing the Square

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

$$76+x\left(112.6-x\right)=x^{2}$$

Use the distributive property to multiply x by 112.6-x.

$$76+112.6x-x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$76+112.6x-x^{2}-x^{2}=0$$

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

$$76+112.6x-2x^{2}=0$$

Subtract 76 from both sides. Anything subtracted from zero gives its negation.

$$112.6x-2x^{2}=-76$$

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.

$$-2x^{2}+112.6x=-76$$

Divide both sides by -2.

$$\frac{-2x^{2}+112.6x}{-2}=-\frac{76}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{112.6}{-2}x=-\frac{76}{-2}$$

Divide 112.6 by -2.

$$x^{2}-56.3x=-\frac{76}{-2}$$

Divide -76 by -2.

$$x^{2}-56.3x=38$$

Divide -56.3, the coefficient of the x term, by 2 to get -28.15. Then add the square of -28.15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-56.3x+\left(-28.15\right)^{2}=38+\left(-28.15\right)^{2}$$

Square -28.15 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-56.3x+792.4225=38+792.4225$$

Add 38 to 792.4225.

$$x^{2}-56.3x+792.4225=830.4225$$

Factor x^{2}-56.3x+792.4225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-28.15\right)^{2}=830.4225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-28.15\right)^{2}}=\sqrt{830.4225}$$

Simplify.

$$x-28.15=\frac{\sqrt{332169}}{20} x-28.15=-\frac{\sqrt{332169}}{20}$$

Add 28.15 to both sides of the equation.

$$x=\frac{\sqrt{332169}+563}{20} x=\frac{563-\sqrt{332169}}{20}$$