Solve (100-7x)(60-5x)=3600

$$( 100 - 7 x ) ( 60 - 5 x ) = 3600$$

Solve for x

$$x = \frac{4 \sqrt{319} + 92}{7} \approx 23.348897771$$

$$x = \frac{92 - 4 \sqrt{319}}{7} \approx 2.936816515$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-7x by 60-5x and combine like terms.

$$6000-920x+35x^{2}=3600$$

Subtract 3600 from both sides.

$$6000-920x+35x^{2}-3600=0$$

Subtract 3600 from 6000 to get 2400.

$$2400-920x+35x^{2}=0$$

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.

$$35x^{2}-920x+2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, -920 for b, and 2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-920\right)±\sqrt{\left(-920\right)^{2}-4\times 35\times 2400}}{2\times 35}$$

Square -920.

$$x=\frac{-\left(-920\right)±\sqrt{846400-4\times 35\times 2400}}{2\times 35}$$

Multiply -4 times 35.

$$x=\frac{-\left(-920\right)±\sqrt{846400-140\times 2400}}{2\times 35}$$

Multiply -140 times 2400.

$$x=\frac{-\left(-920\right)±\sqrt{846400-336000}}{2\times 35}$$

Add 846400 to -336000.

$$x=\frac{-\left(-920\right)±\sqrt{510400}}{2\times 35}$$

Take the square root of 510400.

$$x=\frac{-\left(-920\right)±40\sqrt{319}}{2\times 35}$$

The opposite of -920 is 920.

$$x=\frac{920±40\sqrt{319}}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{920±40\sqrt{319}}{70}$$

Now solve the equation x=\frac{920±40\sqrt{319}}{70} when ± is plus. Add 920 to 40\sqrt{319}.

$$x=\frac{40\sqrt{319}+920}{70}$$

Divide 920+40\sqrt{319} by 70.

$$x=\frac{4\sqrt{319}+92}{7}$$

Now solve the equation x=\frac{920±40\sqrt{319}}{70} when ± is minus. Subtract 40\sqrt{319} from 920.

$$x=\frac{920-40\sqrt{319}}{70}$$

Divide 920-40\sqrt{319} by 70.

$$x=\frac{92-4\sqrt{319}}{7}$$

The equation is now solved.

$$x=\frac{4\sqrt{319}+92}{7} x=\frac{92-4\sqrt{319}}{7}$$

Steps for Completing the Square

Use the distributive property to multiply 100-7x by 60-5x and combine like terms.

$$6000-920x+35x^{2}=3600$$

Subtract 6000 from both sides.

$$-920x+35x^{2}=3600-6000$$

Subtract 6000 from 3600 to get -2400.

$$-920x+35x^{2}=-2400$$

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.

$$35x^{2}-920x=-2400$$

Divide both sides by 35.

$$\frac{35x^{2}-920x}{35}=-\frac{2400}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$x^{2}+\left(-\frac{920}{35}\right)x=-\frac{2400}{35}$$

Reduce the fraction \frac{-920}{35} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{184}{7}x=-\frac{2400}{35}$$

Reduce the fraction \frac{-2400}{35} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{184}{7}x=-\frac{480}{7}$$

Divide -\frac{184}{7}, the coefficient of the x term, by 2 to get -\frac{92}{7}. Then add the square of -\frac{92}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{184}{7}x+\left(-\frac{92}{7}\right)^{2}=-\frac{480}{7}+\left(-\frac{92}{7}\right)^{2}$$

Square -\frac{92}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{184}{7}x+\frac{8464}{49}=-\frac{480}{7}+\frac{8464}{49}$$

Add -\frac{480}{7} to \frac{8464}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{184}{7}x+\frac{8464}{49}=\frac{5104}{49}$$

Factor x^{2}-\frac{184}{7}x+\frac{8464}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{92}{7}\right)^{2}=\frac{5104}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{92}{7}\right)^{2}}=\sqrt{\frac{5104}{49}}$$

Simplify.

$$x-\frac{92}{7}=\frac{4\sqrt{319}}{7} x-\frac{92}{7}=-\frac{4\sqrt{319}}{7}$$

Add \frac{92}{7} to both sides of the equation.

$$x=\frac{4\sqrt{319}+92}{7} x=\frac{92-4\sqrt{319}}{7}$$