Solve 2x(38-2x)=180

$$2x(38-2x)=180$$

Solve for x

$$x = \frac{\sqrt{181} + 19}{2} \approx 16.226812024$$

$$x = \frac{19 - \sqrt{181}}{2} \approx 2.773187976$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 38-2x.

$$76x-4x^{2}=180$$

Subtract 180 from both sides.

$$76x-4x^{2}-180=0$$

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.

$$-4x^{2}+76x-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 76 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-4\right)\left(-180\right)}}{2\left(-4\right)}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-4\right)\left(-180\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-76±\sqrt{5776+16\left(-180\right)}}{2\left(-4\right)}$$

Multiply 16 times -180.

$$x=\frac{-76±\sqrt{5776-2880}}{2\left(-4\right)}$$

Add 5776 to -2880.

$$x=\frac{-76±\sqrt{2896}}{2\left(-4\right)}$$

Take the square root of 2896.

$$x=\frac{-76±4\sqrt{181}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-76±4\sqrt{181}}{-8}$$

Now solve the equation x=\frac{-76±4\sqrt{181}}{-8} when ± is plus. Add -76 to 4\sqrt{181}.

$$x=\frac{4\sqrt{181}-76}{-8}$$

Divide -76+4\sqrt{181} by -8.

$$x=\frac{19-\sqrt{181}}{2}$$

Now solve the equation x=\frac{-76±4\sqrt{181}}{-8} when ± is minus. Subtract 4\sqrt{181} from -76.

$$x=\frac{-4\sqrt{181}-76}{-8}$$

Divide -76-4\sqrt{181} by -8.

$$x=\frac{\sqrt{181}+19}{2}$$

The equation is now solved.

$$x=\frac{19-\sqrt{181}}{2} x=\frac{\sqrt{181}+19}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 38-2x.

$$76x-4x^{2}=180$$

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.

$$-4x^{2}+76x=180$$

Divide both sides by -4.

$$\frac{-4x^{2}+76x}{-4}=\frac{180}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{76}{-4}x=\frac{180}{-4}$$

Divide 76 by -4.

$$x^{2}-19x=\frac{180}{-4}$$

Divide 180 by -4.

$$x^{2}-19x=-45$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-45+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=-45+\frac{361}{4}$$

Add -45 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=\frac{181}{4}$$

Factor x^{2}-19x+\frac{361}{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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{181}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{181}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{\sqrt{181}}{2} x-\frac{19}{2}=-\frac{\sqrt{181}}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=\frac{\sqrt{181}+19}{2} x=\frac{19-\sqrt{181}}{2}$$