Solve 76+x*(1126-x)=x*x

$$76+x \times (1126-x)=x \times x$$

Solve for x

$$x = \frac{\sqrt{317121} + 563}{2} \approx 563.06748747$$

$$x=\frac{563-\sqrt{317121}}{2}\approx -0.06748747$$

Steps Using the Quadratic Formula

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

$$76+x\left(1126-x\right)=x^{2}$$

Use the distributive property to multiply x by 1126-x.

$$76+1126x-x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$76+1126x-x^{2}-x^{2}=0$$

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

$$76+1126x-2x^{2}=0$$

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.

$$-2x^{2}+1126x+76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1126 for b, and 76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1126±\sqrt{1126^{2}-4\left(-2\right)\times 76}}{2\left(-2\right)}$$

Square 1126.

$$x=\frac{-1126±\sqrt{1267876-4\left(-2\right)\times 76}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1126±\sqrt{1267876+8\times 76}}{2\left(-2\right)}$$

Multiply 8 times 76.

$$x=\frac{-1126±\sqrt{1267876+608}}{2\left(-2\right)}$$

Add 1267876 to 608.

$$x=\frac{-1126±\sqrt{1268484}}{2\left(-2\right)}$$

Take the square root of 1268484.

$$x=\frac{-1126±2\sqrt{317121}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1126±2\sqrt{317121}}{-4}$$

Now solve the equation x=\frac{-1126±2\sqrt{317121}}{-4} when ± is plus. Add -1126 to 2\sqrt{317121}.

$$x=\frac{2\sqrt{317121}-1126}{-4}$$

Divide -1126+2\sqrt{317121} by -4.

$$x=\frac{563-\sqrt{317121}}{2}$$

Now solve the equation x=\frac{-1126±2\sqrt{317121}}{-4} when ± is minus. Subtract 2\sqrt{317121} from -1126.

$$x=\frac{-2\sqrt{317121}-1126}{-4}$$

Divide -1126-2\sqrt{317121} by -4.

$$x=\frac{\sqrt{317121}+563}{2}$$

The equation is now solved.

$$x=\frac{563-\sqrt{317121}}{2} x=\frac{\sqrt{317121}+563}{2}$$

Steps for Completing the Square

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

$$76+x\left(1126-x\right)=x^{2}$$

Use the distributive property to multiply x by 1126-x.

$$76+1126x-x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$76+1126x-x^{2}-x^{2}=0$$

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

$$76+1126x-2x^{2}=0$$

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

$$1126x-2x^{2}=-76$$

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.

$$-2x^{2}+1126x=-76$$

Divide both sides by -2.

$$\frac{-2x^{2}+1126x}{-2}=-\frac{76}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1126}{-2}x=-\frac{76}{-2}$$

Divide 1126 by -2.

$$x^{2}-563x=-\frac{76}{-2}$$

Divide -76 by -2.

$$x^{2}-563x=38$$

Divide -563, the coefficient of the x term, by 2 to get -\frac{563}{2}. Then add the square of -\frac{563}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-563x+\left(-\frac{563}{2}\right)^{2}=38+\left(-\frac{563}{2}\right)^{2}$$

Square -\frac{563}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-563x+\frac{316969}{4}=38+\frac{316969}{4}$$

Add 38 to \frac{316969}{4}.

$$x^{2}-563x+\frac{316969}{4}=\frac{317121}{4}$$

Factor x^{2}-563x+\frac{316969}{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}.

$$\left(x-\frac{563}{2}\right)^{2}=\frac{317121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{563}{2}\right)^{2}}=\sqrt{\frac{317121}{4}}$$

Simplify.

$$x-\frac{563}{2}=\frac{\sqrt{317121}}{2} x-\frac{563}{2}=-\frac{\sqrt{317121}}{2}$$

Add \frac{563}{2} to both sides of the equation.

$$x=\frac{\sqrt{317121}+563}{2} x=\frac{563-\sqrt{317121}}{2}$$