Solve 1/22(2x-3)(3x-5)=44

$$\frac { 1 } { 2 } 2 ( 2 x - 3 ) ( 3 x - 5 ) = 44$$

Solve for x

$$x = \frac{\sqrt{1057} + 19}{12} \approx 4.292628035$$

$$x=\frac{19-\sqrt{1057}}{12}\approx -1.125961368$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 2 to get 1.

$$\left(2x-3\right)\left(3x-5\right)=44$$

Use the distributive property to multiply 2x-3 by 3x-5 and combine like terms.

$$6x^{2}-19x+15=44$$

Subtract 44 from both sides.

$$6x^{2}-19x+15-44=0$$

Subtract 44 from 15 to get -29.

$$6x^{2}-19x-29=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -19 for b, and -29 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 6\left(-29\right)}}{2\times 6}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 6\left(-29\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-19\right)±\sqrt{361-24\left(-29\right)}}{2\times 6}$$

Multiply -24 times -29.

$$x=\frac{-\left(-19\right)±\sqrt{361+696}}{2\times 6}$$

Add 361 to 696.

$$x=\frac{-\left(-19\right)±\sqrt{1057}}{2\times 6}$$

The opposite of -19 is 19.

$$x=\frac{19±\sqrt{1057}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{19±\sqrt{1057}}{12}$$

Now solve the equation x=\frac{19±\sqrt{1057}}{12} when ± is plus. Add 19 to \sqrt{1057}.

$$x=\frac{\sqrt{1057}+19}{12}$$

Now solve the equation x=\frac{19±\sqrt{1057}}{12} when ± is minus. Subtract \sqrt{1057} from 19.

$$x=\frac{19-\sqrt{1057}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{1057}+19}{12} x=\frac{19-\sqrt{1057}}{12}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 2 to get 1.

$$\left(2x-3\right)\left(3x-5\right)=44$$

Use the distributive property to multiply 2x-3 by 3x-5 and combine like terms.

$$6x^{2}-19x+15=44$$

Subtract 15 from both sides.

$$6x^{2}-19x=44-15$$

Subtract 15 from 44 to get 29.

$$6x^{2}-19x=29$$

Divide both sides by 6.

$$\frac{6x^{2}-19x}{6}=\frac{29}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{19}{6}x=\frac{29}{6}$$

Divide -\frac{19}{6}, the coefficient of the x term, by 2 to get -\frac{19}{12}. Then add the square of -\frac{19}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{6}x+\left(-\frac{19}{12}\right)^{2}=\frac{29}{6}+\left(-\frac{19}{12}\right)^{2}$$

Square -\frac{19}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{6}x+\frac{361}{144}=\frac{29}{6}+\frac{361}{144}$$

Add \frac{29}{6} to \frac{361}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{6}x+\frac{361}{144}=\frac{1057}{144}$$

Factor x^{2}-\frac{19}{6}x+\frac{361}{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}.

$$\left(x-\frac{19}{12}\right)^{2}=\frac{1057}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{12}\right)^{2}}=\sqrt{\frac{1057}{144}}$$

Simplify.

$$x-\frac{19}{12}=\frac{\sqrt{1057}}{12} x-\frac{19}{12}=-\frac{\sqrt{1057}}{12}$$

Add \frac{19}{12} to both sides of the equation.

$$x=\frac{\sqrt{1057}+19}{12} x=\frac{19-\sqrt{1057}}{12}$$