Solve (x-5)(x+2)div3=5div6

$$(x-5)(x+2) \div 3=5 \div 6$$

Solve for x

$$x = \frac{\sqrt{59} + 3}{2} \approx 5.340572874$$

$$x=\frac{3-\sqrt{59}}{2}\approx -2.340572874$$

Steps Using the Quadratic Formula

Multiply both sides by 3.

$$\left(x-5\right)\left(x+2\right)=\frac{5}{6}\times 3$$

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

$$x^{2}-3x-10=\frac{5}{6}\times 3$$

Multiply \frac{5}{6} and 3 to get \frac{5}{2}.

$$x^{2}-3x-10=\frac{5}{2}$$

Subtract \frac{5}{2} from both sides.

$$x^{2}-3x-10-\frac{5}{2}=0$$

Subtract \frac{5}{2} from -10 to get -\frac{25}{2}.

$$x^{2}-3x-\frac{25}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -\frac{25}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-\frac{25}{2}\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-\frac{25}{2}\right)}}{2}$$

Multiply -4 times -\frac{25}{2}.

$$x=\frac{-\left(-3\right)±\sqrt{9+50}}{2}$$

Add 9 to 50.

$$x=\frac{-\left(-3\right)±\sqrt{59}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{59}}{2}$$

Now solve the equation x=\frac{3±\sqrt{59}}{2} when ± is plus. Add 3 to \sqrt{59}.

$$x=\frac{\sqrt{59}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{59}}{2} when ± is minus. Subtract \sqrt{59} from 3.

$$x=\frac{3-\sqrt{59}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{59}+3}{2} x=\frac{3-\sqrt{59}}{2}$$

Steps for Completing the Square

Multiply both sides by 3.

$$\left(x-5\right)\left(x+2\right)=\frac{5}{6}\times 3$$

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

$$x^{2}-3x-10=\frac{5}{6}\times 3$$

Multiply \frac{5}{6} and 3 to get \frac{5}{2}.

$$x^{2}-3x-10=\frac{5}{2}$$

Add 10 to both sides.

$$x^{2}-3x=\frac{5}{2}+10$$

Add \frac{5}{2} and 10 to get \frac{25}{2}.

$$x^{2}-3x=\frac{25}{2}$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=\frac{25}{2}+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{25}{2}+\frac{9}{4}$$

Add \frac{25}{2} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=\frac{59}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{59}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{59}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{59}}{2} x-\frac{3}{2}=-\frac{\sqrt{59}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{59}+3}{2} x=\frac{3-\sqrt{59}}{2}$$