Solve (2x-3)(3x-2)=21

$$( 2 x - 3 ) ( 3 x - 2 ) = 21$$

Solve for x

$$x=-\frac{5}{6}\approx -0.833333333$$

$$x=3$$

Steps Using the Quadratic Formula

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

$$6x^{2}-13x+6=21$$

Subtract 21 from both sides.

$$6x^{2}-13x+6-21=0$$

Subtract 21 from 6 to get -15.

$$6x^{2}-13x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -13 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\left(-15\right)}}{2\times 6}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 6\left(-15\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-13\right)±\sqrt{169-24\left(-15\right)}}{2\times 6}$$

Multiply -24 times -15.

$$x=\frac{-\left(-13\right)±\sqrt{169+360}}{2\times 6}$$

Add 169 to 360.

$$x=\frac{-\left(-13\right)±\sqrt{529}}{2\times 6}$$

Take the square root of 529.

$$x=\frac{-\left(-13\right)±23}{2\times 6}$$

The opposite of -13 is 13.

$$x=\frac{13±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{13±23}{12}$$

Now solve the equation x=\frac{13±23}{12} when ± is plus. Add 13 to 23.

$$x=\frac{36}{12}$$

Divide 36 by 12.

$$x=3$$

Now solve the equation x=\frac{13±23}{12} when ± is minus. Subtract 23 from 13.

$$x=-\frac{10}{12}$$

Reduce the fraction \frac{-10}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{6}$$

The equation is now solved.

$$x=3 x=-\frac{5}{6}$$

Steps for Completing the Square

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

$$6x^{2}-13x+6=21$$

Subtract 6 from both sides.

$$6x^{2}-13x=21-6$$

Subtract 6 from 21 to get 15.

$$6x^{2}-13x=15$$

Divide both sides by 6.

$$\frac{6x^{2}-13x}{6}=\frac{15}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{13}{6}x=\frac{15}{6}$$

Reduce the fraction \frac{15}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{13}{6}x=\frac{5}{2}$$

Divide -\frac{13}{6}, the coefficient of the x term, by 2 to get -\frac{13}{12}. Then add the square of -\frac{13}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=\frac{5}{2}+\left(-\frac{13}{12}\right)^{2}$$

Square -\frac{13}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{5}{2}+\frac{169}{144}$$

Add \frac{5}{2} to \frac{169}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{529}{144}$$

Factor x^{2}-\frac{13}{6}x+\frac{169}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x-\frac{13}{12}=\frac{23}{12} x-\frac{13}{12}=-\frac{23}{12}$$

Add \frac{13}{12} to both sides of the equation.

$$x=3 x=-\frac{5}{6}$$