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

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

Solve for x

$$x=76$$

$$x=112.6$$

Steps Using the Quadratic Formula

Multiply 112.6 and 76 to get 8557.6.

$$8557.6=\left(76+112.6-x\right)x$$

Add 76 and 112.6 to get 188.6.

$$8557.6=\left(188.6-x\right)x$$

Use the distributive property to multiply 188.6-x by x.

$$8557.6=188.6x-x^{2}$$

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

$$188.6x-x^{2}=8557.6$$

Subtract 8557.6 from both sides.

$$188.6x-x^{2}-8557.6=0$$

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.

$$-x^{2}+188.6x-8557.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 188.6 for b, and -8557.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-188.6±\sqrt{188.6^{2}-4\left(-1\right)\left(-8557.6\right)}}{2\left(-1\right)}$$

Square 188.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-188.6±\sqrt{35569.96-4\left(-1\right)\left(-8557.6\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-188.6±\sqrt{35569.96+4\left(-8557.6\right)}}{2\left(-1\right)}$$

Multiply 4 times -8557.6.

$$x=\frac{-188.6±\sqrt{35569.96-34230.4}}{2\left(-1\right)}$$

Add 35569.96 to -34230.4 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-188.6±\sqrt{1339.56}}{2\left(-1\right)}$$

Take the square root of 1339.56.

$$x=\frac{-188.6±\frac{183}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-188.6±\frac{183}{5}}{-2}$$

Now solve the equation x=\frac{-188.6±\frac{183}{5}}{-2} when ± is plus. Add -188.6 to \frac{183}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{152}{-2}$$

Divide -152 by -2.

$$x=76$$

Now solve the equation x=\frac{-188.6±\frac{183}{5}}{-2} when ± is minus. Subtract \frac{183}{5} from -188.6 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{1126}{5}}{-2}$$

Divide -\frac{1126}{5} by -2.

$$x=\frac{563}{5}$$

The equation is now solved.

$$x=76 x=\frac{563}{5}$$

Steps for Completing the Square

Multiply 112.6 and 76 to get 8557.6.

$$8557.6=\left(76+112.6-x\right)x$$

Add 76 and 112.6 to get 188.6.

$$8557.6=\left(188.6-x\right)x$$

Use the distributive property to multiply 188.6-x by x.

$$8557.6=188.6x-x^{2}$$

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

$$188.6x-x^{2}=8557.6$$

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.

$$-x^{2}+188.6x=8557.6$$

Divide both sides by -1.

$$\frac{-x^{2}+188.6x}{-1}=\frac{8557.6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{188.6}{-1}x=\frac{8557.6}{-1}$$

Divide 188.6 by -1.

$$x^{2}-188.6x=\frac{8557.6}{-1}$$

Divide 8557.6 by -1.

$$x^{2}-188.6x=-8557.6$$

Divide -188.6, the coefficient of the x term, by 2 to get -94.3. Then add the square of -94.3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-188.6x+\left(-94.3\right)^{2}=-8557.6+\left(-94.3\right)^{2}$$

Square -94.3 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-188.6x+8892.49=-8557.6+8892.49$$

Add -8557.6 to 8892.49 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-188.6x+8892.49=334.89$$

Factor x^{2}-188.6x+8892.49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-94.3\right)^{2}=334.89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-94.3\right)^{2}}=\sqrt{334.89}$$

Simplify.

$$x-94.3=\frac{183}{10} x-94.3=-\frac{183}{10}$$

Add 94.3 to both sides of the equation.

$$x=\frac{563}{5} x=76$$