Solve (x-15)(2x-42)=90

$$(x-15)(2x-42)=90$$

Solve for x

$$x=3\sqrt{6}+18\approx 25.348469228$$

$$x=18-3\sqrt{6}\approx 10.651530772$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-15 by 2x-42 and combine like terms.

$$2x^{2}-72x+630=90$$

Subtract 90 from both sides.

$$2x^{2}-72x+630-90=0$$

Subtract 90 from 630 to get 540.

$$2x^{2}-72x+540=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -72 for b, and 540 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 2\times 540}}{2\times 2}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 2\times 540}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-72\right)±\sqrt{5184-8\times 540}}{2\times 2}$$

Multiply -8 times 540.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4320}}{2\times 2}$$

Add 5184 to -4320.

$$x=\frac{-\left(-72\right)±\sqrt{864}}{2\times 2}$$

Take the square root of 864.

$$x=\frac{-\left(-72\right)±12\sqrt{6}}{2\times 2}$$

The opposite of -72 is 72.

$$x=\frac{72±12\sqrt{6}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{72±12\sqrt{6}}{4}$$

Now solve the equation x=\frac{72±12\sqrt{6}}{4} when ± is plus. Add 72 to 12\sqrt{6}.

$$x=\frac{12\sqrt{6}+72}{4}$$

Divide 72+12\sqrt{6} by 4.

$$x=3\sqrt{6}+18$$

Now solve the equation x=\frac{72±12\sqrt{6}}{4} when ± is minus. Subtract 12\sqrt{6} from 72.

$$x=\frac{72-12\sqrt{6}}{4}$$

Divide 72-12\sqrt{6} by 4.

$$x=18-3\sqrt{6}$$

The equation is now solved.

$$x=3\sqrt{6}+18 x=18-3\sqrt{6}$$

Steps for Completing the Square

Use the distributive property to multiply x-15 by 2x-42 and combine like terms.

$$2x^{2}-72x+630=90$$

Subtract 630 from both sides.

$$2x^{2}-72x=90-630$$

Subtract 630 from 90 to get -540.

$$2x^{2}-72x=-540$$

Divide both sides by 2.

$$\frac{2x^{2}-72x}{2}=-\frac{540}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{72}{2}\right)x=-\frac{540}{2}$$

Divide -72 by 2.

$$x^{2}-36x=-\frac{540}{2}$$

Divide -540 by 2.

$$x^{2}-36x=-270$$

Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-36x+\left(-18\right)^{2}=-270+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-270+324$$

Add -270 to 324.

$$x^{2}-36x+324=54$$

Factor x^{2}-36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-18\right)^{2}=54$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{54}$$

Simplify.

$$x-18=3\sqrt{6} x-18=-3\sqrt{6}$$

Add 18 to both sides of the equation.

$$x=3\sqrt{6}+18 x=18-3\sqrt{6}$$