Solve 2x(91+2-2x)=1080

$$2x(91+2-2x)=1080$$

Solve for x

$$x = \frac{3 \sqrt{481} + 93}{4} \approx 39.69878415$$

$$x = \frac{93 - 3 \sqrt{481}}{4} \approx 6.80121585$$

Steps Using the Quadratic Formula

Add 91 and 2 to get 93.

$$2x\left(93-2x\right)=1080$$

Use the distributive property to multiply 2x by 93-2x.

$$186x-4x^{2}=1080$$

Subtract 1080 from both sides.

$$186x-4x^{2}-1080=0$$

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.

$$-4x^{2}+186x-1080=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 186 for b, and -1080 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-186±\sqrt{186^{2}-4\left(-4\right)\left(-1080\right)}}{2\left(-4\right)}$$

Square 186.

$$x=\frac{-186±\sqrt{34596-4\left(-4\right)\left(-1080\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-186±\sqrt{34596+16\left(-1080\right)}}{2\left(-4\right)}$$

Multiply 16 times -1080.

$$x=\frac{-186±\sqrt{34596-17280}}{2\left(-4\right)}$$

Add 34596 to -17280.

$$x=\frac{-186±\sqrt{17316}}{2\left(-4\right)}$$

Take the square root of 17316.

$$x=\frac{-186±6\sqrt{481}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-186±6\sqrt{481}}{-8}$$

Now solve the equation x=\frac{-186±6\sqrt{481}}{-8} when ± is plus. Add -186 to 6\sqrt{481}.

$$x=\frac{6\sqrt{481}-186}{-8}$$

Divide -186+6\sqrt{481} by -8.

$$x=\frac{93-3\sqrt{481}}{4}$$

Now solve the equation x=\frac{-186±6\sqrt{481}}{-8} when ± is minus. Subtract 6\sqrt{481} from -186.

$$x=\frac{-6\sqrt{481}-186}{-8}$$

Divide -186-6\sqrt{481} by -8.

$$x=\frac{3\sqrt{481}+93}{4}$$

The equation is now solved.

$$x=\frac{93-3\sqrt{481}}{4} x=\frac{3\sqrt{481}+93}{4}$$

Steps for Completing the Square

Add 91 and 2 to get 93.

$$2x\left(93-2x\right)=1080$$

Use the distributive property to multiply 2x by 93-2x.

$$186x-4x^{2}=1080$$

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.

$$-4x^{2}+186x=1080$$

Divide both sides by -4.

$$\frac{-4x^{2}+186x}{-4}=\frac{1080}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{186}{-4}x=\frac{1080}{-4}$$

Reduce the fraction \frac{186}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{93}{2}x=\frac{1080}{-4}$$

Divide 1080 by -4.

$$x^{2}-\frac{93}{2}x=-270$$

Divide -\frac{93}{2}, the coefficient of the x term, by 2 to get -\frac{93}{4}. Then add the square of -\frac{93}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{93}{2}x+\left(-\frac{93}{4}\right)^{2}=-270+\left(-\frac{93}{4}\right)^{2}$$

Square -\frac{93}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{93}{2}x+\frac{8649}{16}=-270+\frac{8649}{16}$$

Add -270 to \frac{8649}{16}.

$$x^{2}-\frac{93}{2}x+\frac{8649}{16}=\frac{4329}{16}$$

Factor x^{2}-\frac{93}{2}x+\frac{8649}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{93}{4}\right)^{2}=\frac{4329}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{93}{4}\right)^{2}}=\sqrt{\frac{4329}{16}}$$

Simplify.

$$x-\frac{93}{4}=\frac{3\sqrt{481}}{4} x-\frac{93}{4}=-\frac{3\sqrt{481}}{4}$$

Add \frac{93}{4} to both sides of the equation.

$$x=\frac{3\sqrt{481}+93}{4} x=\frac{93-3\sqrt{481}}{4}$$