Solve (6x+5)(x-3)=-2x(7x+5)+x-12

$$( 6 x + 5 ) ( x - 3 ) = - 2 x ( 7 x + 5 ) + x - 12$$

Solve for x

$$x=-\frac{3}{10}=-0.3$$

$$x=\frac{1}{2}=0.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x+5 by x-3 and combine like terms.

$$6x^{2}-13x-15=-2x\left(7x+5\right)+x-12$$

Use the distributive property to multiply -2x by 7x+5.

$$6x^{2}-13x-15=-14x^{2}-10x+x-12$$

Combine -10x and x to get -9x.

$$6x^{2}-13x-15=-14x^{2}-9x-12$$

Add 14x^{2} to both sides.

$$6x^{2}-13x-15+14x^{2}=-9x-12$$

Combine 6x^{2} and 14x^{2} to get 20x^{2}.

$$20x^{2}-13x-15=-9x-12$$

Add 9x to both sides.

$$20x^{2}-13x-15+9x=-12$$

Combine -13x and 9x to get -4x.

$$20x^{2}-4x-15=-12$$

Add 12 to both sides.

$$20x^{2}-4x-15+12=0$$

Add -15 and 12 to get -3.

$$20x^{2}-4x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -4 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 20\left(-3\right)}}{2\times 20}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 20\left(-3\right)}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-4\right)±\sqrt{16-80\left(-3\right)}}{2\times 20}$$

Multiply -80 times -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+240}}{2\times 20}$$

Add 16 to 240.

$$x=\frac{-\left(-4\right)±\sqrt{256}}{2\times 20}$$

Take the square root of 256.

$$x=\frac{-\left(-4\right)±16}{2\times 20}$$

The opposite of -4 is 4.

$$x=\frac{4±16}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{4±16}{40}$$

Now solve the equation x=\frac{4±16}{40} when ± is plus. Add 4 to 16.

$$x=\frac{20}{40}$$

Reduce the fraction \frac{20}{40} to lowest terms by extracting and canceling out 20.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{4±16}{40} when ± is minus. Subtract 16 from 4.

$$x=-\frac{12}{40}$$

Reduce the fraction \frac{-12}{40} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{10}$$

The equation is now solved.

$$x=\frac{1}{2} x=-\frac{3}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 6x+5 by x-3 and combine like terms.

$$6x^{2}-13x-15=-2x\left(7x+5\right)+x-12$$

Use the distributive property to multiply -2x by 7x+5.

$$6x^{2}-13x-15=-14x^{2}-10x+x-12$$

Combine -10x and x to get -9x.

$$6x^{2}-13x-15=-14x^{2}-9x-12$$

Add 14x^{2} to both sides.

$$6x^{2}-13x-15+14x^{2}=-9x-12$$

Combine 6x^{2} and 14x^{2} to get 20x^{2}.

$$20x^{2}-13x-15=-9x-12$$

Add 9x to both sides.

$$20x^{2}-13x-15+9x=-12$$

Combine -13x and 9x to get -4x.

$$20x^{2}-4x-15=-12$$

Add 15 to both sides.

$$20x^{2}-4x=-12+15$$

Add -12 and 15 to get 3.

$$20x^{2}-4x=3$$

Divide both sides by 20.

$$\frac{20x^{2}-4x}{20}=\frac{3}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}+\left(-\frac{4}{20}\right)x=\frac{3}{20}$$

Reduce the fraction \frac{-4}{20} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{1}{5}x=\frac{3}{20}$$

Divide -\frac{1}{5}, the coefficient of the x term, by 2 to get -\frac{1}{10}. Then add the square of -\frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{5}x+\left(-\frac{1}{10}\right)^{2}=\frac{3}{20}+\left(-\frac{1}{10}\right)^{2}$$

Square -\frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{3}{20}+\frac{1}{100}$$

Add \frac{3}{20} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{4}{25}$$

Factor x^{2}-\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{10}\right)^{2}=\frac{4}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{10}\right)^{2}}=\sqrt{\frac{4}{25}}$$

Simplify.

$$x-\frac{1}{10}=\frac{2}{5} x-\frac{1}{10}=-\frac{2}{5}$$

Add \frac{1}{10} to both sides of the equation.

$$x=\frac{1}{2} x=-\frac{3}{10}$$