Solve (x-2)(x-5)=7

$$( x - 2 ) ( x - 5 ) = 7$$

Solve for x

$$x = \frac{\sqrt{37} + 7}{2} \approx 6.541381265$$

$$x=\frac{7-\sqrt{37}}{2}\approx 0.458618735$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2 by x-5 and combine like terms.

$$x^{2}-7x+10=7$$

Subtract 7 from both sides.

$$x^{2}-7x+10-7=0$$

Subtract 7 from 10 to get 3.

$$x^{2}-7x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12}}{2}$$

Add 49 to -12.

$$x=\frac{-\left(-7\right)±\sqrt{37}}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{37}}{2}$$

Now solve the equation x=\frac{7±\sqrt{37}}{2} when ± is plus. Add 7 to \sqrt{37}.

$$x=\frac{\sqrt{37}+7}{2}$$

Now solve the equation x=\frac{7±\sqrt{37}}{2} when ± is minus. Subtract \sqrt{37} from 7.

$$x=\frac{7-\sqrt{37}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{37}+7}{2} x=\frac{7-\sqrt{37}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-2 by x-5 and combine like terms.

$$x^{2}-7x+10=7$$

Subtract 10 from both sides.

$$x^{2}-7x=7-10$$

Subtract 10 from 7 to get -3.

$$x^{2}-7x=-3$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-3+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-3+\frac{49}{4}$$

Add -3 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{37}{4}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{37}}{2} x-\frac{7}{2}=-\frac{\sqrt{37}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{\sqrt{37}+7}{2} x=\frac{7-\sqrt{37}}{2}$$