Solve (35-x)(75-x)=2100

$$( 35 - x ) ( 75 - x ) = 2100$$

Solve for x

$$x=105$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 35-x by 75-x and combine like terms.

$$2625-110x+x^{2}=2100$$

Subtract 2100 from both sides.

$$2625-110x+x^{2}-2100=0$$

Subtract 2100 from 2625 to get 525.

$$525-110x+x^{2}=0$$

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.

$$x^{2}-110x+525=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -110 for b, and 525 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\times 525}}{2}$$

Square -110.

$$x=\frac{-\left(-110\right)±\sqrt{12100-4\times 525}}{2}$$

Multiply -4 times 525.

$$x=\frac{-\left(-110\right)±\sqrt{12100-2100}}{2}$$

Add 12100 to -2100.

$$x=\frac{-\left(-110\right)±\sqrt{10000}}{2}$$

Take the square root of 10000.

$$x=\frac{-\left(-110\right)±100}{2}$$

The opposite of -110 is 110.

$$x=\frac{110±100}{2}$$

Now solve the equation x=\frac{110±100}{2} when ± is plus. Add 110 to 100.

$$x=\frac{210}{2}$$

Divide 210 by 2.

$$x=105$$

Now solve the equation x=\frac{110±100}{2} when ± is minus. Subtract 100 from 110.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

The equation is now solved.

$$x=105 x=5$$

Steps for Completing the Square

Use the distributive property to multiply 35-x by 75-x and combine like terms.

$$2625-110x+x^{2}=2100$$

Subtract 2625 from both sides.

$$-110x+x^{2}=2100-2625$$

Subtract 2625 from 2100 to get -525.

$$-110x+x^{2}=-525$$

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.

$$x^{2}-110x=-525$$

Divide -110, the coefficient of the x term, by 2 to get -55. Then add the square of -55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-110x+\left(-55\right)^{2}=-525+\left(-55\right)^{2}$$

Square -55.

$$x^{2}-110x+3025=-525+3025$$

Add -525 to 3025.

$$x^{2}-110x+3025=2500$$

Factor x^{2}-110x+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-55\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-55=50 x-55=-50$$

Add 55 to both sides of the equation.

$$x=105 x=5$$