Solve (40-3x)*(30-2x)=500

$$(40-3x) \times (30-2x)=500$$

Solve for x

$$x=5$$

$$x = \frac{70}{3} = 23\frac{1}{3} \approx 23.333333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-3x by 30-2x and combine like terms.

$$1200-170x+6x^{2}=500$$

Subtract 500 from both sides.

$$1200-170x+6x^{2}-500=0$$

Subtract 500 from 1200 to get 700.

$$700-170x+6x^{2}=0$$

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.

$$6x^{2}-170x+700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -170 for b, and 700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-170\right)±\sqrt{\left(-170\right)^{2}-4\times 6\times 700}}{2\times 6}$$

Square -170.

$$x=\frac{-\left(-170\right)±\sqrt{28900-4\times 6\times 700}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-170\right)±\sqrt{28900-24\times 700}}{2\times 6}$$

Multiply -24 times 700.

$$x=\frac{-\left(-170\right)±\sqrt{28900-16800}}{2\times 6}$$

Add 28900 to -16800.

$$x=\frac{-\left(-170\right)±\sqrt{12100}}{2\times 6}$$

Take the square root of 12100.

$$x=\frac{-\left(-170\right)±110}{2\times 6}$$

The opposite of -170 is 170.

$$x=\frac{170±110}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{170±110}{12}$$

Now solve the equation x=\frac{170±110}{12} when ± is plus. Add 170 to 110.

$$x=\frac{280}{12}$$

Reduce the fraction \frac{280}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{70}{3}$$

Now solve the equation x=\frac{170±110}{12} when ± is minus. Subtract 110 from 170.

$$x=\frac{60}{12}$$

Divide 60 by 12.

$$x=5$$

The equation is now solved.

$$x=\frac{70}{3} x=5$$

Steps for Completing the Square

Use the distributive property to multiply 40-3x by 30-2x and combine like terms.

$$1200-170x+6x^{2}=500$$

Subtract 1200 from both sides.

$$-170x+6x^{2}=500-1200$$

Subtract 1200 from 500 to get -700.

$$-170x+6x^{2}=-700$$

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.

$$6x^{2}-170x=-700$$

Divide both sides by 6.

$$\frac{6x^{2}-170x}{6}=-\frac{700}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{170}{6}\right)x=-\frac{700}{6}$$

Reduce the fraction \frac{-170}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{85}{3}x=-\frac{700}{6}$$

Reduce the fraction \frac{-700}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{85}{3}x=-\frac{350}{3}$$

Divide -\frac{85}{3}, the coefficient of the x term, by 2 to get -\frac{85}{6}. Then add the square of -\frac{85}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{85}{3}x+\left(-\frac{85}{6}\right)^{2}=-\frac{350}{3}+\left(-\frac{85}{6}\right)^{2}$$

Square -\frac{85}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{85}{3}x+\frac{7225}{36}=-\frac{350}{3}+\frac{7225}{36}$$

Add -\frac{350}{3} to \frac{7225}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{85}{3}x+\frac{7225}{36}=\frac{3025}{36}$$

Factor x^{2}-\frac{85}{3}x+\frac{7225}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{85}{6}\right)^{2}=\frac{3025}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{6}\right)^{2}}=\sqrt{\frac{3025}{36}}$$

Simplify.

$$x-\frac{85}{6}=\frac{55}{6} x-\frac{85}{6}=-\frac{55}{6}$$

Add \frac{85}{6} to both sides of the equation.

$$x=\frac{70}{3} x=5$$