Solve (x-3)(2x+3)=x+6

$$( x - 3 ) ( 2 x + 3 ) = x + 6$$

Solve for x

$$x=\frac{\sqrt{34}}{2}+1\approx 3.915475947$$

$$x=-\frac{\sqrt{34}}{2}+1\approx -1.915475947$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-3 by 2x+3 and combine like terms.

$$2x^{2}-3x-9=x+6$$

Subtract x from both sides.

$$2x^{2}-3x-9-x=6$$

Combine -3x and -x to get -4x.

$$2x^{2}-4x-9=6$$

Subtract 6 from both sides.

$$2x^{2}-4x-9-6=0$$

Subtract 6 from -9 to get -15.

$$2x^{2}-4x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-15\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-15\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-15\right)}}{2\times 2}$$

Multiply -8 times -15.

$$x=\frac{-\left(-4\right)±\sqrt{16+120}}{2\times 2}$$

Add 16 to 120.

$$x=\frac{-\left(-4\right)±\sqrt{136}}{2\times 2}$$

Take the square root of 136.

$$x=\frac{-\left(-4\right)±2\sqrt{34}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{34}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±2\sqrt{34}}{4}$$

Now solve the equation x=\frac{4±2\sqrt{34}}{4} when ± is plus. Add 4 to 2\sqrt{34}.

$$x=\frac{2\sqrt{34}+4}{4}$$

Divide 4+2\sqrt{34} by 4.

$$x=\frac{\sqrt{34}}{2}+1$$

Now solve the equation x=\frac{4±2\sqrt{34}}{4} when ± is minus. Subtract 2\sqrt{34} from 4.

$$x=\frac{4-2\sqrt{34}}{4}$$

Divide 4-2\sqrt{34} by 4.

$$x=-\frac{\sqrt{34}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{34}}{2}+1 x=-\frac{\sqrt{34}}{2}+1$$

Steps for Completing the Square

Use the distributive property to multiply x-3 by 2x+3 and combine like terms.

$$2x^{2}-3x-9=x+6$$

Subtract x from both sides.

$$2x^{2}-3x-9-x=6$$

Combine -3x and -x to get -4x.

$$2x^{2}-4x-9=6$$

Add 9 to both sides.

$$2x^{2}-4x=6+9$$

Add 6 and 9 to get 15.

$$2x^{2}-4x=15$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{15}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{15}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{15}{2}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{15}{2}+1$$

Add \frac{15}{2} to 1.

$$x^{2}-2x+1=\frac{17}{2}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{17}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{17}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{34}}{2} x-1=-\frac{\sqrt{34}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{34}}{2}+1 x=-\frac{\sqrt{34}}{2}+1$$