Solve 6x^2+6x-1=5x-4

$$6 x ^ { 2 } + 6 x - 1 = 5 x - 4$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{71}i}{12}\approx -0.083333333+0.702179148i$$

$$x=\frac{-\sqrt{71}i-1}{12}\approx -0.083333333-0.702179148i$$

Steps Using the Quadratic Formula

Subtract 5x from both sides.

$$6x^{2}+6x-1-5x=-4$$

Combine 6x and -5x to get x.

$$6x^{2}+x-1=-4$$

Add 4 to both sides.

$$6x^{2}+x-1+4=0$$

Add -1 and 4 to get 3.

$$6x^{2}+x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 1 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 6\times 3}}{2\times 6}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 6\times 3}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-1±\sqrt{1-24\times 3}}{2\times 6}$$

Multiply -24 times 3.

$$x=\frac{-1±\sqrt{1-72}}{2\times 6}$$

Add 1 to -72.

$$x=\frac{-1±\sqrt{-71}}{2\times 6}$$

Take the square root of -71.

$$x=\frac{-1±\sqrt{71}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-1±\sqrt{71}i}{12}$$

Now solve the equation x=\frac{-1±\sqrt{71}i}{12} when ± is plus. Add -1 to i\sqrt{71}.

$$x=\frac{-1+\sqrt{71}i}{12}$$

Now solve the equation x=\frac{-1±\sqrt{71}i}{12} when ± is minus. Subtract i\sqrt{71} from -1.

$$x=\frac{-\sqrt{71}i-1}{12}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{71}i}{12} x=\frac{-\sqrt{71}i-1}{12}$$

Steps for Completing the Square

Subtract 5x from both sides.

$$6x^{2}+6x-1-5x=-4$$

Combine 6x and -5x to get x.

$$6x^{2}+x-1=-4$$

Add 1 to both sides.

$$6x^{2}+x=-4+1$$

Add -4 and 1 to get -3.

$$6x^{2}+x=-3$$

Divide both sides by 6.

$$\frac{6x^{2}+x}{6}=-\frac{3}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{1}{6}x=-\frac{3}{6}$$

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

$$x^{2}+\frac{1}{6}x=-\frac{1}{2}$$

Divide \frac{1}{6}, the coefficient of the x term, by 2 to get \frac{1}{12}. Then add the square of \frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{6}x+\left(\frac{1}{12}\right)^{2}=-\frac{1}{2}+\left(\frac{1}{12}\right)^{2}$$

Square \frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=-\frac{1}{2}+\frac{1}{144}$$

Add -\frac{1}{2} to \frac{1}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=-\frac{71}{144}$$

Factor x^{2}+\frac{1}{6}x+\frac{1}{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}.

$$\left(x+\frac{1}{12}\right)^{2}=-\frac{71}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{12}\right)^{2}}=\sqrt{-\frac{71}{144}}$$

Simplify.

$$x+\frac{1}{12}=\frac{\sqrt{71}i}{12} x+\frac{1}{12}=-\frac{\sqrt{71}i}{12}$$

Subtract \frac{1}{12} from both sides of the equation.

$$x=\frac{-1+\sqrt{71}i}{12} x=\frac{-\sqrt{71}i-1}{12}$$