Solve left(40-xright)(30-x)=1140

$$\left( 40-x \right) (30-x) = 1140$$

Solve for x

$$x=35-\sqrt{1165}\approx 0.867903668$$

$$x=\sqrt{1165}+35\approx 69.132096332$$

Steps Using the Quadratic Formula

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

$$1200-70x+x^{2}=1140$$

Subtract 1140 from both sides.

$$1200-70x+x^{2}-1140=0$$

Subtract 1140 from 1200 to get 60.

$$60-70x+x^{2}=0$$

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.

$$x^{2}-70x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -70 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 60}}{2}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 60}}{2}$$

Multiply -4 times 60.

$$x=\frac{-\left(-70\right)±\sqrt{4900-240}}{2}$$

Add 4900 to -240.

$$x=\frac{-\left(-70\right)±\sqrt{4660}}{2}$$

Take the square root of 4660.

$$x=\frac{-\left(-70\right)±2\sqrt{1165}}{2}$$

The opposite of -70 is 70.

$$x=\frac{70±2\sqrt{1165}}{2}$$

Now solve the equation x=\frac{70±2\sqrt{1165}}{2} when ± is plus. Add 70 to 2\sqrt{1165}.

$$x=\frac{2\sqrt{1165}+70}{2}$$

Divide 70+2\sqrt{1165} by 2.

$$x=\sqrt{1165}+35$$

Now solve the equation x=\frac{70±2\sqrt{1165}}{2} when ± is minus. Subtract 2\sqrt{1165} from 70.

$$x=\frac{70-2\sqrt{1165}}{2}$$

Divide 70-2\sqrt{1165} by 2.

$$x=35-\sqrt{1165}$$

The equation is now solved.

$$x=\sqrt{1165}+35 x=35-\sqrt{1165}$$

Steps for Completing the Square

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

$$1200-70x+x^{2}=1140$$

Subtract 1200 from both sides.

$$-70x+x^{2}=1140-1200$$

Subtract 1200 from 1140 to get -60.

$$-70x+x^{2}=-60$$

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.

$$x^{2}-70x=-60$$

Divide -70, the coefficient of the x term, by 2 to get -35. Then add the square of -35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-70x+\left(-35\right)^{2}=-60+\left(-35\right)^{2}$$

Square -35.

$$x^{2}-70x+1225=-60+1225$$

Add -60 to 1225.

$$x^{2}-70x+1225=1165$$

Factor x^{2}-70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-35\right)^{2}=1165$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-35\right)^{2}}=\sqrt{1165}$$

Simplify.

$$x-35=\sqrt{1165} x-35=-\sqrt{1165}$$

Add 35 to both sides of the equation.

$$x=\sqrt{1165}+35 x=35-\sqrt{1165}$$