Solve (15-x)*(400+5x)=4250

$$(15-x) \times (400+5x)=4250$$

Solve for x

$$x=-70$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 15-x by 400+5x and combine like terms.

$$6000-325x-5x^{2}=4250$$

Subtract 4250 from both sides.

$$6000-325x-5x^{2}-4250=0$$

Subtract 4250 from 6000 to get 1750.

$$1750-325x-5x^{2}=0$$

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.

$$-5x^{2}-325x+1750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -325 for b, and 1750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-325\right)±\sqrt{\left(-325\right)^{2}-4\left(-5\right)\times 1750}}{2\left(-5\right)}$$

Square -325.

$$x=\frac{-\left(-325\right)±\sqrt{105625-4\left(-5\right)\times 1750}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-325\right)±\sqrt{105625+20\times 1750}}{2\left(-5\right)}$$

Multiply 20 times 1750.

$$x=\frac{-\left(-325\right)±\sqrt{105625+35000}}{2\left(-5\right)}$$

Add 105625 to 35000.

$$x=\frac{-\left(-325\right)±\sqrt{140625}}{2\left(-5\right)}$$

Take the square root of 140625.

$$x=\frac{-\left(-325\right)±375}{2\left(-5\right)}$$

The opposite of -325 is 325.

$$x=\frac{325±375}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{325±375}{-10}$$

Now solve the equation x=\frac{325±375}{-10} when ± is plus. Add 325 to 375.

$$x=\frac{700}{-10}$$

Divide 700 by -10.

$$x=-70$$

Now solve the equation x=\frac{325±375}{-10} when ± is minus. Subtract 375 from 325.

$$x=-\frac{50}{-10}$$

Divide -50 by -10.

$$x=5$$

The equation is now solved.

$$x=-70 x=5$$

Steps for Completing the Square

Use the distributive property to multiply 15-x by 400+5x and combine like terms.

$$6000-325x-5x^{2}=4250$$

Subtract 6000 from both sides.

$$-325x-5x^{2}=4250-6000$$

Subtract 6000 from 4250 to get -1750.

$$-325x-5x^{2}=-1750$$

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.

$$-5x^{2}-325x=-1750$$

Divide both sides by -5.

$$\frac{-5x^{2}-325x}{-5}=-\frac{1750}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{325}{-5}\right)x=-\frac{1750}{-5}$$

Divide -325 by -5.

$$x^{2}+65x=-\frac{1750}{-5}$$

Divide -1750 by -5.

$$x^{2}+65x=350$$

Divide 65, the coefficient of the x term, by 2 to get \frac{65}{2}. Then add the square of \frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+65x+\left(\frac{65}{2}\right)^{2}=350+\left(\frac{65}{2}\right)^{2}$$

Square \frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+65x+\frac{4225}{4}=350+\frac{4225}{4}$$

Add 350 to \frac{4225}{4}.

$$x^{2}+65x+\frac{4225}{4}=\frac{5625}{4}$$

Factor x^{2}+65x+\frac{4225}{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}.

$$\left(x+\frac{65}{2}\right)^{2}=\frac{5625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{65}{2}\right)^{2}}=\sqrt{\frac{5625}{4}}$$

Simplify.

$$x+\frac{65}{2}=\frac{75}{2} x+\frac{65}{2}=-\frac{75}{2}$$

Subtract \frac{65}{2} from both sides of the equation.

$$x=5 x=-70$$