Solve (15-x)(5x+500)=4250

$$( 15 - x ) ( 5 x + 500 ) = 4250$$

Solve for x

$$x = \frac{5 \sqrt{393} - 85}{2} \approx 7.060569004$$

$$x=\frac{-5\sqrt{393}-85}{2}\approx -92.060569004$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 15-x by 5x+500 and combine like terms.

$$-425x+7500-5x^{2}=4250$$

Subtract 4250 from both sides.

$$-425x+7500-5x^{2}-4250=0$$

Subtract 4250 from 7500 to get 3250.

$$-425x+3250-5x^{2}=0$$

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.

$$-5x^{2}-425x+3250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -425 for b, and 3250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-425\right)±\sqrt{\left(-425\right)^{2}-4\left(-5\right)\times 3250}}{2\left(-5\right)}$$

Square -425.

$$x=\frac{-\left(-425\right)±\sqrt{180625-4\left(-5\right)\times 3250}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-425\right)±\sqrt{180625+20\times 3250}}{2\left(-5\right)}$$

Multiply 20 times 3250.

$$x=\frac{-\left(-425\right)±\sqrt{180625+65000}}{2\left(-5\right)}$$

Add 180625 to 65000.

$$x=\frac{-\left(-425\right)±\sqrt{245625}}{2\left(-5\right)}$$

Take the square root of 245625.

$$x=\frac{-\left(-425\right)±25\sqrt{393}}{2\left(-5\right)}$$

The opposite of -425 is 425.

$$x=\frac{425±25\sqrt{393}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{425±25\sqrt{393}}{-10}$$

Now solve the equation x=\frac{425±25\sqrt{393}}{-10} when ± is plus. Add 425 to 25\sqrt{393}.

$$x=\frac{25\sqrt{393}+425}{-10}$$

Divide 425+25\sqrt{393} by -10.

$$x=\frac{-5\sqrt{393}-85}{2}$$

Now solve the equation x=\frac{425±25\sqrt{393}}{-10} when ± is minus. Subtract 25\sqrt{393} from 425.

$$x=\frac{425-25\sqrt{393}}{-10}$$

Divide 425-25\sqrt{393} by -10.

$$x=\frac{5\sqrt{393}-85}{2}$$

The equation is now solved.

$$x=\frac{-5\sqrt{393}-85}{2} x=\frac{5\sqrt{393}-85}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 15-x by 5x+500 and combine like terms.

$$-425x+7500-5x^{2}=4250$$

Subtract 7500 from both sides.

$$-425x-5x^{2}=4250-7500$$

Subtract 7500 from 4250 to get -3250.

$$-425x-5x^{2}=-3250$$

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.

$$-5x^{2}-425x=-3250$$

Divide both sides by -5.

$$\frac{-5x^{2}-425x}{-5}=-\frac{3250}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{425}{-5}\right)x=-\frac{3250}{-5}$$

Divide -425 by -5.

$$x^{2}+85x=-\frac{3250}{-5}$$

Divide -3250 by -5.

$$x^{2}+85x=650$$

Divide 85, the coefficient of the x term, by 2 to get \frac{85}{2}. Then add the square of \frac{85}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+85x+\left(\frac{85}{2}\right)^{2}=650+\left(\frac{85}{2}\right)^{2}$$

Square \frac{85}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+85x+\frac{7225}{4}=650+\frac{7225}{4}$$

Add 650 to \frac{7225}{4}.

$$x^{2}+85x+\frac{7225}{4}=\frac{9825}{4}$$

Factor x^{2}+85x+\frac{7225}{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}.

$$\left(x+\frac{85}{2}\right)^{2}=\frac{9825}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{85}{2}\right)^{2}}=\sqrt{\frac{9825}{4}}$$

Simplify.

$$x+\frac{85}{2}=\frac{5\sqrt{393}}{2} x+\frac{85}{2}=-\frac{5\sqrt{393}}{2}$$

Subtract \frac{85}{2} from both sides of the equation.

$$x=\frac{5\sqrt{393}-85}{2} x=\frac{-5\sqrt{393}-85}{2}$$