Solve (175-x)x=4000

$$(175-x)x=4000$$

Solve for x

$$x = \frac{15 \sqrt{65} + 175}{2} \approx 147.966933112$$

$$x = \frac{175 - 15 \sqrt{65}}{2} \approx 27.033066888$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 175-x by x.

$$175x-x^{2}=4000$$

Subtract 4000 from both sides.

$$175x-x^{2}-4000=0$$

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.

$$-x^{2}+175x-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 175 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-175±\sqrt{175^{2}-4\left(-1\right)\left(-4000\right)}}{2\left(-1\right)}$$

Square 175.

$$x=\frac{-175±\sqrt{30625-4\left(-1\right)\left(-4000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-175±\sqrt{30625+4\left(-4000\right)}}{2\left(-1\right)}$$

Multiply 4 times -4000.

$$x=\frac{-175±\sqrt{30625-16000}}{2\left(-1\right)}$$

Add 30625 to -16000.

$$x=\frac{-175±\sqrt{14625}}{2\left(-1\right)}$$

Take the square root of 14625.

$$x=\frac{-175±15\sqrt{65}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-175±15\sqrt{65}}{-2}$$

Now solve the equation x=\frac{-175±15\sqrt{65}}{-2} when ± is plus. Add -175 to 15\sqrt{65}.

$$x=\frac{15\sqrt{65}-175}{-2}$$

Divide -175+15\sqrt{65} by -2.

$$x=\frac{175-15\sqrt{65}}{2}$$

Now solve the equation x=\frac{-175±15\sqrt{65}}{-2} when ± is minus. Subtract 15\sqrt{65} from -175.

$$x=\frac{-15\sqrt{65}-175}{-2}$$

Divide -175-15\sqrt{65} by -2.

$$x=\frac{15\sqrt{65}+175}{2}$$

The equation is now solved.

$$x=\frac{175-15\sqrt{65}}{2} x=\frac{15\sqrt{65}+175}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 175-x by x.

$$175x-x^{2}=4000$$

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.

$$-x^{2}+175x=4000$$

Divide both sides by -1.

$$\frac{-x^{2}+175x}{-1}=\frac{4000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{175}{-1}x=\frac{4000}{-1}$$

Divide 175 by -1.

$$x^{2}-175x=\frac{4000}{-1}$$

Divide 4000 by -1.

$$x^{2}-175x=-4000$$

Divide -175, the coefficient of the x term, by 2 to get -\frac{175}{2}. Then add the square of -\frac{175}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-175x+\left(-\frac{175}{2}\right)^{2}=-4000+\left(-\frac{175}{2}\right)^{2}$$

Square -\frac{175}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-175x+\frac{30625}{4}=-4000+\frac{30625}{4}$$

Add -4000 to \frac{30625}{4}.

$$x^{2}-175x+\frac{30625}{4}=\frac{14625}{4}$$

Factor x^{2}-175x+\frac{30625}{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}.

$$\left(x-\frac{175}{2}\right)^{2}=\frac{14625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{175}{2}\right)^{2}}=\sqrt{\frac{14625}{4}}$$

Simplify.

$$x-\frac{175}{2}=\frac{15\sqrt{65}}{2} x-\frac{175}{2}=-\frac{15\sqrt{65}}{2}$$

Add \frac{175}{2} to both sides of the equation.

$$x=\frac{15\sqrt{65}+175}{2} x=\frac{175-15\sqrt{65}}{2}$$