Solve (4-2x)(26-x)=86x

$$( 4 - 2 x ) ( 26 - x ) = 86 x$$

Solve for x

$$x = \frac{3 \sqrt{537} + 71}{2} \approx 70.259890679$$

$$x=\frac{71-3\sqrt{537}}{2}\approx 0.740109321$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4-2x by 26-x and combine like terms.

$$104-56x+2x^{2}=86x$$

Subtract 86x from both sides.

$$104-56x+2x^{2}-86x=0$$

Combine -56x and -86x to get -142x.

$$104-142x+2x^{2}=0$$

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.

$$2x^{2}-142x+104=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -142 for b, and 104 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-142\right)±\sqrt{\left(-142\right)^{2}-4\times 2\times 104}}{2\times 2}$$

Square -142.

$$x=\frac{-\left(-142\right)±\sqrt{20164-4\times 2\times 104}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-142\right)±\sqrt{20164-8\times 104}}{2\times 2}$$

Multiply -8 times 104.

$$x=\frac{-\left(-142\right)±\sqrt{20164-832}}{2\times 2}$$

Add 20164 to -832.

$$x=\frac{-\left(-142\right)±\sqrt{19332}}{2\times 2}$$

Take the square root of 19332.

$$x=\frac{-\left(-142\right)±6\sqrt{537}}{2\times 2}$$

The opposite of -142 is 142.

$$x=\frac{142±6\sqrt{537}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{142±6\sqrt{537}}{4}$$

Now solve the equation x=\frac{142±6\sqrt{537}}{4} when ± is plus. Add 142 to 6\sqrt{537}.

$$x=\frac{6\sqrt{537}+142}{4}$$

Divide 142+6\sqrt{537} by 4.

$$x=\frac{3\sqrt{537}+71}{2}$$

Now solve the equation x=\frac{142±6\sqrt{537}}{4} when ± is minus. Subtract 6\sqrt{537} from 142.

$$x=\frac{142-6\sqrt{537}}{4}$$

Divide 142-6\sqrt{537} by 4.

$$x=\frac{71-3\sqrt{537}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{537}+71}{2} x=\frac{71-3\sqrt{537}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 4-2x by 26-x and combine like terms.

$$104-56x+2x^{2}=86x$$

Subtract 86x from both sides.

$$104-56x+2x^{2}-86x=0$$

Combine -56x and -86x to get -142x.

$$104-142x+2x^{2}=0$$

Subtract 104 from both sides. Anything subtracted from zero gives its negation.

$$-142x+2x^{2}=-104$$

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.

$$2x^{2}-142x=-104$$

Divide both sides by 2.

$$\frac{2x^{2}-142x}{2}=-\frac{104}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{142}{2}\right)x=-\frac{104}{2}$$

Divide -142 by 2.

$$x^{2}-71x=-\frac{104}{2}$$

Divide -104 by 2.

$$x^{2}-71x=-52$$

Divide -71, the coefficient of the x term, by 2 to get -\frac{71}{2}. Then add the square of -\frac{71}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-71x+\left(-\frac{71}{2}\right)^{2}=-52+\left(-\frac{71}{2}\right)^{2}$$

Square -\frac{71}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-71x+\frac{5041}{4}=-52+\frac{5041}{4}$$

Add -52 to \frac{5041}{4}.

$$x^{2}-71x+\frac{5041}{4}=\frac{4833}{4}$$

Factor x^{2}-71x+\frac{5041}{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}.

$$\left(x-\frac{71}{2}\right)^{2}=\frac{4833}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{71}{2}\right)^{2}}=\sqrt{\frac{4833}{4}}$$

Simplify.

$$x-\frac{71}{2}=\frac{3\sqrt{537}}{2} x-\frac{71}{2}=-\frac{3\sqrt{537}}{2}$$

Add \frac{71}{2} to both sides of the equation.

$$x=\frac{3\sqrt{537}+71}{2} x=\frac{71-3\sqrt{537}}{2}$$