Solve (2x+1)=(x-3)(3x+5)

$$( 2 x + 1 ) = ( x - 3 ) ( 3 x + 5 )$$

Solve for x

$$x=\frac{\sqrt{57}}{3}+1\approx 3.516611478$$

$$x=-\frac{\sqrt{57}}{3}+1\approx -1.516611478$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-3 by 3x+5 and combine like terms.

$$2x+1=3x^{2}-4x-15$$

Subtract 3x^{2} from both sides.

$$2x+1-3x^{2}=-4x-15$$

Add 4x to both sides.

$$2x+1-3x^{2}+4x=-15$$

Combine 2x and 4x to get 6x.

$$6x+1-3x^{2}=-15$$

Add 15 to both sides.

$$6x+1-3x^{2}+15=0$$

Add 1 and 15 to get 16.

$$6x+16-3x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-3x^{2}+6x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 6 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-6±\sqrt{36+12\times 16}}{2\left(-3\right)}$$

Multiply 12 times 16.

$$x=\frac{-6±\sqrt{36+192}}{2\left(-3\right)}$$

Add 36 to 192.

$$x=\frac{-6±\sqrt{228}}{2\left(-3\right)}$$

Take the square root of 228.

$$x=\frac{-6±2\sqrt{57}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-6±2\sqrt{57}}{-6}$$

Now solve the equation x=\frac{-6±2\sqrt{57}}{-6} when ± is plus. Add -6 to 2\sqrt{57}.

$$x=\frac{2\sqrt{57}-6}{-6}$$

Divide -6+2\sqrt{57} by -6.

$$x=-\frac{\sqrt{57}}{3}+1$$

Now solve the equation x=\frac{-6±2\sqrt{57}}{-6} when ± is minus. Subtract 2\sqrt{57} from -6.

$$x=\frac{-2\sqrt{57}-6}{-6}$$

Divide -6-2\sqrt{57} by -6.

$$x=\frac{\sqrt{57}}{3}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{57}}{3}+1 x=\frac{\sqrt{57}}{3}+1$$

Steps for Completing the Square

Use the distributive property to multiply x-3 by 3x+5 and combine like terms.

$$2x+1=3x^{2}-4x-15$$

Subtract 3x^{2} from both sides.

$$2x+1-3x^{2}=-4x-15$$

Add 4x to both sides.

$$2x+1-3x^{2}+4x=-15$$

Combine 2x and 4x to get 6x.

$$6x+1-3x^{2}=-15$$

Subtract 1 from both sides.

$$6x-3x^{2}=-15-1$$

Subtract 1 from -15 to get -16.

$$6x-3x^{2}=-16$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-3x^{2}+6x=-16$$

Divide both sides by -3.

$$\frac{-3x^{2}+6x}{-3}=-\frac{16}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{6}{-3}x=-\frac{16}{-3}$$

Divide 6 by -3.

$$x^{2}-2x=-\frac{16}{-3}$$

Divide -16 by -3.

$$x^{2}-2x=\frac{16}{3}$$

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.

$$x^{2}-2x+1=\frac{16}{3}+1$$

Add \frac{16}{3} to 1.

$$x^{2}-2x+1=\frac{19}{3}$$

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}.

$$\left(x-1\right)^{2}=\frac{19}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{19}{3}}$$

Simplify.

$$x-1=\frac{\sqrt{57}}{3} x-1=-\frac{\sqrt{57}}{3}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{57}}{3}+1 x=-\frac{\sqrt{57}}{3}+1$$