Solve (x-14)*(x-30)=200

$$(x-14) \times (x-30)=200$$

Solve for x

$$x=2\sqrt{66}+22\approx 38.248076809$$

$$x=22-2\sqrt{66}\approx 5.751923191$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-14 by x-30 and combine like terms.

$$x^{2}-44x+420=200$$

Subtract 200 from both sides.

$$x^{2}-44x+420-200=0$$

Subtract 200 from 420 to get 220.

$$x^{2}-44x+220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -44 for b, and 220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 220}}{2}$$

Square -44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-4\times 220}}{2}$$

Multiply -4 times 220.

$$x=\frac{-\left(-44\right)±\sqrt{1936-880}}{2}$$

Add 1936 to -880.

$$x=\frac{-\left(-44\right)±\sqrt{1056}}{2}$$

Take the square root of 1056.

$$x=\frac{-\left(-44\right)±4\sqrt{66}}{2}$$

The opposite of -44 is 44.

$$x=\frac{44±4\sqrt{66}}{2}$$

Now solve the equation x=\frac{44±4\sqrt{66}}{2} when ± is plus. Add 44 to 4\sqrt{66}.

$$x=\frac{4\sqrt{66}+44}{2}$$

Divide 44+4\sqrt{66} by 2.

$$x=2\sqrt{66}+22$$

Now solve the equation x=\frac{44±4\sqrt{66}}{2} when ± is minus. Subtract 4\sqrt{66} from 44.

$$x=\frac{44-4\sqrt{66}}{2}$$

Divide 44-4\sqrt{66} by 2.

$$x=22-2\sqrt{66}$$

The equation is now solved.

$$x=2\sqrt{66}+22 x=22-2\sqrt{66}$$

Steps for Completing the Square

Use the distributive property to multiply x-14 by x-30 and combine like terms.

$$x^{2}-44x+420=200$$

Subtract 420 from both sides.

$$x^{2}-44x=200-420$$

Subtract 420 from 200 to get -220.

$$x^{2}-44x=-220$$

Divide -44, the coefficient of the x term, by 2 to get -22. Then add the square of -22 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-44x+\left(-22\right)^{2}=-220+\left(-22\right)^{2}$$

Square -22.

$$x^{2}-44x+484=-220+484$$

Add -220 to 484.

$$x^{2}-44x+484=264$$

Factor x^{2}-44x+484. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-22\right)^{2}=264$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-22\right)^{2}}=\sqrt{264}$$

Simplify.

$$x-22=2\sqrt{66} x-22=-2\sqrt{66}$$

Add 22 to both sides of the equation.

$$x=2\sqrt{66}+22 x=22-2\sqrt{66}$$