Solve 6x^2+11x-10=4x-15

$$6 x ^ { 2 } + 11 x - 10 = 4 x - 15$$

Solve for x (complex solution)

$$x=\frac{-7+\sqrt{71}i}{12}\approx -0.583333333+0.702179148i$$

$$x=\frac{-\sqrt{71}i-7}{12}\approx -0.583333333-0.702179148i$$

Steps Using the Quadratic Formula

Subtract 4x from both sides.

$$6x^{2}+11x-10-4x=-15$$

Combine 11x and -4x to get 7x.

$$6x^{2}+7x-10=-15$$

Add 15 to both sides.

$$6x^{2}+7x-10+15=0$$

Add -10 and 15 to get 5.

$$6x^{2}+7x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 7 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\times 5}}{2\times 6}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 6\times 5}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-7±\sqrt{49-24\times 5}}{2\times 6}$$

Multiply -24 times 5.

$$x=\frac{-7±\sqrt{49-120}}{2\times 6}$$

Add 49 to -120.

$$x=\frac{-7±\sqrt{-71}}{2\times 6}$$

Take the square root of -71.

$$x=\frac{-7±\sqrt{71}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-7±\sqrt{71}i}{12}$$

Now solve the equation x=\frac{-7±\sqrt{71}i}{12} when ± is plus. Add -7 to i\sqrt{71}.

$$x=\frac{-7+\sqrt{71}i}{12}$$

Now solve the equation x=\frac{-7±\sqrt{71}i}{12} when ± is minus. Subtract i\sqrt{71} from -7.

$$x=\frac{-\sqrt{71}i-7}{12}$$

The equation is now solved.

$$x=\frac{-7+\sqrt{71}i}{12} x=\frac{-\sqrt{71}i-7}{12}$$

Steps for Completing the Square

Subtract 4x from both sides.

$$6x^{2}+11x-10-4x=-15$$

Combine 11x and -4x to get 7x.

$$6x^{2}+7x-10=-15$$

Add 10 to both sides.

$$6x^{2}+7x=-15+10$$

Add -15 and 10 to get -5.

$$6x^{2}+7x=-5$$

Divide both sides by 6.

$$\frac{6x^{2}+7x}{6}=-\frac{5}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{7}{6}x=-\frac{5}{6}$$

Divide \frac{7}{6}, the coefficient of the x term, by 2 to get \frac{7}{12}. Then add the square of \frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=-\frac{5}{6}+\frac{49}{144}$$

Add -\frac{5}{6} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=-\frac{71}{144}$$

Factor x^{2}+\frac{7}{6}x+\frac{49}{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}.

$$\left(x+\frac{7}{12}\right)^{2}=-\frac{71}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{-\frac{71}{144}}$$

Simplify.

$$x+\frac{7}{12}=\frac{\sqrt{71}i}{12} x+\frac{7}{12}=-\frac{\sqrt{71}i}{12}$$

Subtract \frac{7}{12} from both sides of the equation.

$$x=\frac{-7+\sqrt{71}i}{12} x=\frac{-\sqrt{71}i-7}{12}$$