Solve (50-x)x=325

$$( 50 - x ) x = 325$$

Solve for x

$$x=10\sqrt{3}+25\approx 42.320508076$$

$$x=25-10\sqrt{3}\approx 7.679491924$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 50-x by x.

$$50x-x^{2}=325$$

Subtract 325 from both sides.

$$50x-x^{2}-325=0$$

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.

$$-x^{2}+50x-325=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 50 for b, and -325 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-1\right)\left(-325\right)}}{2\left(-1\right)}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-1\right)\left(-325\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-50±\sqrt{2500+4\left(-325\right)}}{2\left(-1\right)}$$

Multiply 4 times -325.

$$x=\frac{-50±\sqrt{2500-1300}}{2\left(-1\right)}$$

Add 2500 to -1300.

$$x=\frac{-50±\sqrt{1200}}{2\left(-1\right)}$$

Take the square root of 1200.

$$x=\frac{-50±20\sqrt{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-50±20\sqrt{3}}{-2}$$

Now solve the equation x=\frac{-50±20\sqrt{3}}{-2} when ± is plus. Add -50 to 20\sqrt{3}.

$$x=\frac{20\sqrt{3}-50}{-2}$$

Divide -50+20\sqrt{3} by -2.

$$x=25-10\sqrt{3}$$

Now solve the equation x=\frac{-50±20\sqrt{3}}{-2} when ± is minus. Subtract 20\sqrt{3} from -50.

$$x=\frac{-20\sqrt{3}-50}{-2}$$

Divide -50-20\sqrt{3} by -2.

$$x=10\sqrt{3}+25$$

The equation is now solved.

$$x=25-10\sqrt{3} x=10\sqrt{3}+25$$

Steps for Completing the Square

Use the distributive property to multiply 50-x by x.

$$50x-x^{2}=325$$

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.

$$-x^{2}+50x=325$$

Divide both sides by -1.

$$\frac{-x^{2}+50x}{-1}=\frac{325}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{50}{-1}x=\frac{325}{-1}$$

Divide 50 by -1.

$$x^{2}-50x=\frac{325}{-1}$$

Divide 325 by -1.

$$x^{2}-50x=-325$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-325+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-325+625$$

Add -325 to 625.

$$x^{2}-50x+625=300$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{300}$$

Simplify.

$$x-25=10\sqrt{3} x-25=-10\sqrt{3}$$

Add 25 to both sides of the equation.

$$x=10\sqrt{3}+25 x=25-10\sqrt{3}$$