Solve (60-4x)(40-3x)=1625

$$( 60 - 4 x ) ( 40 - 3 x ) = 1625$$

Solve for x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$x = \frac{155}{6} = 25\frac{5}{6} \approx 25.833333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 60-4x by 40-3x and combine like terms.

$$2400-340x+12x^{2}=1625$$

Subtract 1625 from both sides.

$$2400-340x+12x^{2}-1625=0$$

Subtract 1625 from 2400 to get 775.

$$775-340x+12x^{2}=0$$

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.

$$12x^{2}-340x+775=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -340 for b, and 775 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-340\right)±\sqrt{\left(-340\right)^{2}-4\times 12\times 775}}{2\times 12}$$

Square -340.

$$x=\frac{-\left(-340\right)±\sqrt{115600-4\times 12\times 775}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-340\right)±\sqrt{115600-48\times 775}}{2\times 12}$$

Multiply -48 times 775.

$$x=\frac{-\left(-340\right)±\sqrt{115600-37200}}{2\times 12}$$

Add 115600 to -37200.

$$x=\frac{-\left(-340\right)±\sqrt{78400}}{2\times 12}$$

Take the square root of 78400.

$$x=\frac{-\left(-340\right)±280}{2\times 12}$$

The opposite of -340 is 340.

$$x=\frac{340±280}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{340±280}{24}$$

Now solve the equation x=\frac{340±280}{24} when ± is plus. Add 340 to 280.

$$x=\frac{620}{24}$$

Reduce the fraction \frac{620}{24} to lowest terms by extracting and canceling out 4.

$$x=\frac{155}{6}$$

Now solve the equation x=\frac{340±280}{24} when ± is minus. Subtract 280 from 340.

$$x=\frac{60}{24}$$

Reduce the fraction \frac{60}{24} to lowest terms by extracting and canceling out 12.

$$x=\frac{5}{2}$$

The equation is now solved.

$$x=\frac{155}{6} x=\frac{5}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 60-4x by 40-3x and combine like terms.

$$2400-340x+12x^{2}=1625$$

Subtract 2400 from both sides.

$$-340x+12x^{2}=1625-2400$$

Subtract 2400 from 1625 to get -775.

$$-340x+12x^{2}=-775$$

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.

$$12x^{2}-340x=-775$$

Divide both sides by 12.

$$\frac{12x^{2}-340x}{12}=-\frac{775}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{340}{12}\right)x=-\frac{775}{12}$$

Reduce the fraction \frac{-340}{12} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{85}{3}x=-\frac{775}{12}$$

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.

$$x^{2}-\frac{85}{3}x+\left(-\frac{85}{6}\right)^{2}=-\frac{775}{12}+\left(-\frac{85}{6}\right)^{2}$$

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

$$x^{2}-\frac{85}{3}x+\frac{7225}{36}=-\frac{775}{12}+\frac{7225}{36}$$

Add -\frac{775}{12} to \frac{7225}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{85}{3}x+\frac{7225}{36}=\frac{1225}{9}$$

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}.

$$\left(x-\frac{85}{6}\right)^{2}=\frac{1225}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{6}\right)^{2}}=\sqrt{\frac{1225}{9}}$$

Simplify.

$$x-\frac{85}{6}=\frac{35}{3} x-\frac{85}{6}=-\frac{35}{3}$$

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

$$x=\frac{155}{6} x=\frac{5}{2}$$