Solve (35-2x)(20-x)=250

$$( 35 - 2 x ) ( 20 - x ) = 250$$

Solve for x

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

$$x=30$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 35-2x by 20-x and combine like terms.

$$700-75x+2x^{2}=250$$

Subtract 250 from both sides.

$$700-75x+2x^{2}-250=0$$

Subtract 250 from 700 to get 450.

$$450-75x+2x^{2}=0$$

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.

$$2x^{2}-75x+450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -75 for b, and 450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-75\right)±\sqrt{\left(-75\right)^{2}-4\times 2\times 450}}{2\times 2}$$

Square -75.

$$x=\frac{-\left(-75\right)±\sqrt{5625-4\times 2\times 450}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-75\right)±\sqrt{5625-8\times 450}}{2\times 2}$$

Multiply -8 times 450.

$$x=\frac{-\left(-75\right)±\sqrt{5625-3600}}{2\times 2}$$

Add 5625 to -3600.

$$x=\frac{-\left(-75\right)±\sqrt{2025}}{2\times 2}$$

Take the square root of 2025.

$$x=\frac{-\left(-75\right)±45}{2\times 2}$$

The opposite of -75 is 75.

$$x=\frac{75±45}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{75±45}{4}$$

Now solve the equation x=\frac{75±45}{4} when ± is plus. Add 75 to 45.

$$x=\frac{120}{4}$$

Divide 120 by 4.

$$x=30$$

Now solve the equation x=\frac{75±45}{4} when ± is minus. Subtract 45 from 75.

$$x=\frac{30}{4}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{15}{2}$$

The equation is now solved.

$$x=30 x=\frac{15}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 35-2x by 20-x and combine like terms.

$$700-75x+2x^{2}=250$$

Subtract 700 from both sides.

$$-75x+2x^{2}=250-700$$

Subtract 700 from 250 to get -450.

$$-75x+2x^{2}=-450$$

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.

$$2x^{2}-75x=-450$$

Divide both sides by 2.

$$\frac{2x^{2}-75x}{2}=-\frac{450}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{75}{2}x=-\frac{450}{2}$$

Divide -450 by 2.

$$x^{2}-\frac{75}{2}x=-225$$

Divide -\frac{75}{2}, the coefficient of the x term, by 2 to get -\frac{75}{4}. Then add the square of -\frac{75}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{75}{2}x+\left(-\frac{75}{4}\right)^{2}=-225+\left(-\frac{75}{4}\right)^{2}$$

Square -\frac{75}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{75}{2}x+\frac{5625}{16}=-225+\frac{5625}{16}$$

Add -225 to \frac{5625}{16}.

$$x^{2}-\frac{75}{2}x+\frac{5625}{16}=\frac{2025}{16}$$

Factor x^{2}-\frac{75}{2}x+\frac{5625}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{75}{4}\right)^{2}=\frac{2025}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{75}{4}\right)^{2}}=\sqrt{\frac{2025}{16}}$$

Simplify.

$$x-\frac{75}{4}=\frac{45}{4} x-\frac{75}{4}=-\frac{45}{4}$$

Add \frac{75}{4} to both sides of the equation.

$$x=30 x=\frac{15}{2}$$