Solve (30-x)(20-x)=557

$$( 30 - x ) ( 20 - x ) = 557$$

Solve for x

$$x=25-\sqrt{582}\approx 0.875323836$$

$$x=\sqrt{582}+25\approx 49.124676164$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 30-x by 20-x and combine like terms.

$$600-50x+x^{2}=557$$

Subtract 557 from both sides.

$$600-50x+x^{2}-557=0$$

Subtract 557 from 600 to get 43.

$$43-50x+x^{2}=0$$

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.

$$x^{2}-50x+43=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -50 for b, and 43 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 43}}{2}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 43}}{2}$$

Multiply -4 times 43.

$$x=\frac{-\left(-50\right)±\sqrt{2500-172}}{2}$$

Add 2500 to -172.

$$x=\frac{-\left(-50\right)±\sqrt{2328}}{2}$$

Take the square root of 2328.

$$x=\frac{-\left(-50\right)±2\sqrt{582}}{2}$$

The opposite of -50 is 50.

$$x=\frac{50±2\sqrt{582}}{2}$$

Now solve the equation x=\frac{50±2\sqrt{582}}{2} when ± is plus. Add 50 to 2\sqrt{582}.

$$x=\frac{2\sqrt{582}+50}{2}$$

Divide 50+2\sqrt{582} by 2.

$$x=\sqrt{582}+25$$

Now solve the equation x=\frac{50±2\sqrt{582}}{2} when ± is minus. Subtract 2\sqrt{582} from 50.

$$x=\frac{50-2\sqrt{582}}{2}$$

Divide 50-2\sqrt{582} by 2.

$$x=25-\sqrt{582}$$

The equation is now solved.

$$x=\sqrt{582}+25 x=25-\sqrt{582}$$

Steps for Completing the Square

Use the distributive property to multiply 30-x by 20-x and combine like terms.

$$600-50x+x^{2}=557$$

Subtract 600 from both sides.

$$-50x+x^{2}=557-600$$

Subtract 600 from 557 to get -43.

$$-50x+x^{2}=-43$$

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.

$$x^{2}-50x=-43$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-43+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-43+625$$

Add -43 to 625.

$$x^{2}-50x+625=582$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=582$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{582}$$

Simplify.

$$x-25=\sqrt{582} x-25=-\sqrt{582}$$

Add 25 to both sides of the equation.

$$x=\sqrt{582}+25 x=25-\sqrt{582}$$