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

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

Solve for x

$$x = \frac{\sqrt{353} + 21}{4} \approx 9.947073557$$

$$x=\frac{21-\sqrt{353}}{4}\approx 0.552926443$$

Steps Using the Quadratic Formula

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

$$6x^{2}-13x+6=\left(2x+5\right)\left(2x-1\right)$$

Use the distributive property to multiply 2x+5 by 2x-1 and combine like terms.

$$6x^{2}-13x+6=4x^{2}+8x-5$$

Subtract 4x^{2} from both sides.

$$6x^{2}-13x+6-4x^{2}=8x-5$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}-13x+6=8x-5$$

Subtract 8x from both sides.

$$2x^{2}-13x+6-8x=-5$$

Combine -13x and -8x to get -21x.

$$2x^{2}-21x+6=-5$$

Add 5 to both sides.

$$2x^{2}-21x+6+5=0$$

Add 6 and 5 to get 11.

$$2x^{2}-21x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -21 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 2\times 11}}{2\times 2}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 2\times 11}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-21\right)±\sqrt{441-8\times 11}}{2\times 2}$$

Multiply -8 times 11.

$$x=\frac{-\left(-21\right)±\sqrt{441-88}}{2\times 2}$$

Add 441 to -88.

$$x=\frac{-\left(-21\right)±\sqrt{353}}{2\times 2}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{353}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{21±\sqrt{353}}{4}$$

Now solve the equation x=\frac{21±\sqrt{353}}{4} when ± is plus. Add 21 to \sqrt{353}.

$$x=\frac{\sqrt{353}+21}{4}$$

Now solve the equation x=\frac{21±\sqrt{353}}{4} when ± is minus. Subtract \sqrt{353} from 21.

$$x=\frac{21-\sqrt{353}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{353}+21}{4} x=\frac{21-\sqrt{353}}{4}$$

Steps for Completing the Square

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

$$6x^{2}-13x+6=\left(2x+5\right)\left(2x-1\right)$$

Use the distributive property to multiply 2x+5 by 2x-1 and combine like terms.

$$6x^{2}-13x+6=4x^{2}+8x-5$$

Subtract 4x^{2} from both sides.

$$6x^{2}-13x+6-4x^{2}=8x-5$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}-13x+6=8x-5$$

Subtract 8x from both sides.

$$2x^{2}-13x+6-8x=-5$$

Combine -13x and -8x to get -21x.

$$2x^{2}-21x+6=-5$$

Subtract 6 from both sides.

$$2x^{2}-21x=-5-6$$

Subtract 6 from -5 to get -11.

$$2x^{2}-21x=-11$$

Divide both sides by 2.

$$\frac{2x^{2}-21x}{2}=-\frac{11}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{21}{2}x=-\frac{11}{2}$$

Divide -\frac{21}{2}, the coefficient of the x term, by 2 to get -\frac{21}{4}. Then add the square of -\frac{21}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{2}x+\left(-\frac{21}{4}\right)^{2}=-\frac{11}{2}+\left(-\frac{21}{4}\right)^{2}$$

Square -\frac{21}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{2}x+\frac{441}{16}=-\frac{11}{2}+\frac{441}{16}$$

Add -\frac{11}{2} to \frac{441}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{21}{2}x+\frac{441}{16}=\frac{353}{16}$$

Factor x^{2}-\frac{21}{2}x+\frac{441}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{4}\right)^{2}=\frac{353}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{4}\right)^{2}}=\sqrt{\frac{353}{16}}$$

Simplify.

$$x-\frac{21}{4}=\frac{\sqrt{353}}{4} x-\frac{21}{4}=-\frac{\sqrt{353}}{4}$$

Add \frac{21}{4} to both sides of the equation.

$$x=\frac{\sqrt{353}+21}{4} x=\frac{21-\sqrt{353}}{4}$$