Solve left(40-2xright)left(26-xright)=86x

$$\left( 40-2x \right) \left( 26-x \right) = 86x$$

Solve for x

$$x = \frac{3 \sqrt{649} + 89}{2} \approx 82.713217609$$

$$x = \frac{89 - 3 \sqrt{649}}{2} \approx 6.286782391$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-2x by 26-x and combine like terms.

$$1040-92x+2x^{2}=86x$$

Subtract 86x from both sides.

$$1040-92x+2x^{2}-86x=0$$

Combine -92x and -86x to get -178x.

$$1040-178x+2x^{2}=0$$

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.

$$2x^{2}-178x+1040=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -178 for b, and 1040 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-178\right)±\sqrt{\left(-178\right)^{2}-4\times 2\times 1040}}{2\times 2}$$

Square -178.

$$x=\frac{-\left(-178\right)±\sqrt{31684-4\times 2\times 1040}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-178\right)±\sqrt{31684-8\times 1040}}{2\times 2}$$

Multiply -8 times 1040.

$$x=\frac{-\left(-178\right)±\sqrt{31684-8320}}{2\times 2}$$

Add 31684 to -8320.

$$x=\frac{-\left(-178\right)±\sqrt{23364}}{2\times 2}$$

Take the square root of 23364.

$$x=\frac{-\left(-178\right)±6\sqrt{649}}{2\times 2}$$

The opposite of -178 is 178.

$$x=\frac{178±6\sqrt{649}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{178±6\sqrt{649}}{4}$$

Now solve the equation x=\frac{178±6\sqrt{649}}{4} when ± is plus. Add 178 to 6\sqrt{649}.

$$x=\frac{6\sqrt{649}+178}{4}$$

Divide 178+6\sqrt{649} by 4.

$$x=\frac{3\sqrt{649}+89}{2}$$

Now solve the equation x=\frac{178±6\sqrt{649}}{4} when ± is minus. Subtract 6\sqrt{649} from 178.

$$x=\frac{178-6\sqrt{649}}{4}$$

Divide 178-6\sqrt{649} by 4.

$$x=\frac{89-3\sqrt{649}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{649}+89}{2} x=\frac{89-3\sqrt{649}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 40-2x by 26-x and combine like terms.

$$1040-92x+2x^{2}=86x$$

Subtract 86x from both sides.

$$1040-92x+2x^{2}-86x=0$$

Combine -92x and -86x to get -178x.

$$1040-178x+2x^{2}=0$$

Subtract 1040 from both sides. Anything subtracted from zero gives its negation.

$$-178x+2x^{2}=-1040$$

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.

$$2x^{2}-178x=-1040$$

Divide both sides by 2.

$$\frac{2x^{2}-178x}{2}=-\frac{1040}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{178}{2}\right)x=-\frac{1040}{2}$$

Divide -178 by 2.

$$x^{2}-89x=-\frac{1040}{2}$$

Divide -1040 by 2.

$$x^{2}-89x=-520$$

Divide -89, the coefficient of the x term, by 2 to get -\frac{89}{2}. Then add the square of -\frac{89}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-89x+\left(-\frac{89}{2}\right)^{2}=-520+\left(-\frac{89}{2}\right)^{2}$$

Square -\frac{89}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-89x+\frac{7921}{4}=-520+\frac{7921}{4}$$

Add -520 to \frac{7921}{4}.

$$x^{2}-89x+\frac{7921}{4}=\frac{5841}{4}$$

Factor x^{2}-89x+\frac{7921}{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}.

$$\left(x-\frac{89}{2}\right)^{2}=\frac{5841}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{89}{2}\right)^{2}}=\sqrt{\frac{5841}{4}}$$

Simplify.

$$x-\frac{89}{2}=\frac{3\sqrt{649}}{2} x-\frac{89}{2}=-\frac{3\sqrt{649}}{2}$$

Add \frac{89}{2} to both sides of the equation.

$$x=\frac{3\sqrt{649}+89}{2} x=\frac{89-3\sqrt{649}}{2}$$