Solve (6x-1)(2x+7)-5/2(3-x)=0

$$( 6 x - 1 ) ( 2 x + 7 ) - \frac { 5 } { 2 } ( 3 - x ) = 0$$

Solve for x

$$x=\frac{\sqrt{10009}-85}{48}\approx 0.313437289$$

$$x=\frac{-\sqrt{10009}-85}{48}\approx -3.855103956$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x-1 by 2x+7 and combine like terms.

$$12x^{2}+40x-7-\frac{5}{2}\left(3-x\right)=0$$

Use the distributive property to multiply -\frac{5}{2} by 3-x.

$$12x^{2}+40x-7-\frac{15}{2}+\frac{5}{2}x=0$$

Subtract \frac{15}{2} from -7 to get -\frac{29}{2}.

$$12x^{2}+40x-\frac{29}{2}+\frac{5}{2}x=0$$

Combine 40x and \frac{5}{2}x to get \frac{85}{2}x.

$$12x^{2}+\frac{85}{2}x-\frac{29}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, \frac{85}{2} for b, and -\frac{29}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{85}{2}±\sqrt{\left(\frac{85}{2}\right)^{2}-4\times 12\left(-\frac{29}{2}\right)}}{2\times 12}$$

Square \frac{85}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{85}{2}±\sqrt{\frac{7225}{4}-4\times 12\left(-\frac{29}{2}\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\frac{85}{2}±\sqrt{\frac{7225}{4}-48\left(-\frac{29}{2}\right)}}{2\times 12}$$

Multiply -48 times -\frac{29}{2}.

$$x=\frac{-\frac{85}{2}±\sqrt{\frac{7225}{4}+696}}{2\times 12}$$

Add \frac{7225}{4} to 696.

$$x=\frac{-\frac{85}{2}±\sqrt{\frac{10009}{4}}}{2\times 12}$$

Take the square root of \frac{10009}{4}.

$$x=\frac{-\frac{85}{2}±\frac{\sqrt{10009}}{2}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-\frac{85}{2}±\frac{\sqrt{10009}}{2}}{24}$$

Now solve the equation x=\frac{-\frac{85}{2}±\frac{\sqrt{10009}}{2}}{24} when ± is plus. Add -\frac{85}{2} to \frac{\sqrt{10009}}{2}.

$$x=\frac{\sqrt{10009}-85}{2\times 24}$$

Divide \frac{-85+\sqrt{10009}}{2} by 24.

$$x=\frac{\sqrt{10009}-85}{48}$$

Now solve the equation x=\frac{-\frac{85}{2}±\frac{\sqrt{10009}}{2}}{24} when ± is minus. Subtract \frac{\sqrt{10009}}{2} from -\frac{85}{2}.

$$x=\frac{-\sqrt{10009}-85}{2\times 24}$$

Divide \frac{-85-\sqrt{10009}}{2} by 24.

$$x=\frac{-\sqrt{10009}-85}{48}$$

The equation is now solved.

$$x=\frac{\sqrt{10009}-85}{48} x=\frac{-\sqrt{10009}-85}{48}$$

Steps for Completing the Square

Use the distributive property to multiply 6x-1 by 2x+7 and combine like terms.

$$12x^{2}+40x-7-\frac{5}{2}\left(3-x\right)=0$$

Use the distributive property to multiply -\frac{5}{2} by 3-x.

$$12x^{2}+40x-7-\frac{15}{2}+\frac{5}{2}x=0$$

Subtract \frac{15}{2} from -7 to get -\frac{29}{2}.

$$12x^{2}+40x-\frac{29}{2}+\frac{5}{2}x=0$$

Combine 40x and \frac{5}{2}x to get \frac{85}{2}x.

$$12x^{2}+\frac{85}{2}x-\frac{29}{2}=0$$

Add \frac{29}{2} to both sides. Anything plus zero gives itself.

$$12x^{2}+\frac{85}{2}x=\frac{29}{2}$$

Divide both sides by 12.

$$\frac{12x^{2}+\frac{85}{2}x}{12}=\frac{\frac{29}{2}}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{\frac{85}{2}}{12}x=\frac{\frac{29}{2}}{12}$$

Divide \frac{85}{2} by 12.

$$x^{2}+\frac{85}{24}x=\frac{\frac{29}{2}}{12}$$

Divide \frac{29}{2} by 12.

$$x^{2}+\frac{85}{24}x=\frac{29}{24}$$

Divide \frac{85}{24}, the coefficient of the x term, by 2 to get \frac{85}{48}. Then add the square of \frac{85}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{85}{24}x+\left(\frac{85}{48}\right)^{2}=\frac{29}{24}+\left(\frac{85}{48}\right)^{2}$$

Square \frac{85}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{85}{24}x+\frac{7225}{2304}=\frac{29}{24}+\frac{7225}{2304}$$

Add \frac{29}{24} to \frac{7225}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{85}{24}x+\frac{7225}{2304}=\frac{10009}{2304}$$

Factor x^{2}+\frac{85}{24}x+\frac{7225}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{85}{48}\right)^{2}=\frac{10009}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{85}{48}\right)^{2}}=\sqrt{\frac{10009}{2304}}$$

Simplify.

$$x+\frac{85}{48}=\frac{\sqrt{10009}}{48} x+\frac{85}{48}=-\frac{\sqrt{10009}}{48}$$

Subtract \frac{85}{48} from both sides of the equation.

$$x=\frac{\sqrt{10009}-85}{48} x=\frac{-\sqrt{10009}-85}{48}$$