Solve (-4)=(12x^2-10x+1)-1

$$( - 4 ) = ( 12 x ^ { 2 } - 10 x + 1 ) - 1$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{23}i}{12}\approx 0.416666667+0.399652627i$$

$$x=\frac{-\sqrt{23}i+5}{12}\approx 0.416666667-0.399652627i$$

Steps Using the Quadratic Formula

Subtract 1 from 1 to get 0.

$$-4=12x^{2}-10x$$

Swap sides so that all variable terms are on the left hand side.

$$12x^{2}-10x=-4$$

Add 4 to both sides.

$$12x^{2}-10x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -10 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 12\times 4}}{2\times 12}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 12\times 4}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-10\right)±\sqrt{100-48\times 4}}{2\times 12}$$

Multiply -48 times 4.

$$x=\frac{-\left(-10\right)±\sqrt{100-192}}{2\times 12}$$

Add 100 to -192.

$$x=\frac{-\left(-10\right)±\sqrt{-92}}{2\times 12}$$

Take the square root of -92.

$$x=\frac{-\left(-10\right)±2\sqrt{23}i}{2\times 12}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{23}i}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{10±2\sqrt{23}i}{24}$$

Now solve the equation x=\frac{10±2\sqrt{23}i}{24} when ± is plus. Add 10 to 2i\sqrt{23}.

$$x=\frac{10+2\sqrt{23}i}{24}$$

Divide 10+2i\sqrt{23} by 24.

$$x=\frac{5+\sqrt{23}i}{12}$$

Now solve the equation x=\frac{10±2\sqrt{23}i}{24} when ± is minus. Subtract 2i\sqrt{23} from 10.

$$x=\frac{-2\sqrt{23}i+10}{24}$$

Divide 10-2i\sqrt{23} by 24.

$$x=\frac{-\sqrt{23}i+5}{12}$$

The equation is now solved.

$$x=\frac{5+\sqrt{23}i}{12} x=\frac{-\sqrt{23}i+5}{12}$$

Steps for Completing the Square

Subtract 1 from 1 to get 0.

$$-4=12x^{2}-10x$$

Swap sides so that all variable terms are on the left hand side.

$$12x^{2}-10x=-4$$

Divide both sides by 12.

$$\frac{12x^{2}-10x}{12}=-\frac{4}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{10}{12}\right)x=-\frac{4}{12}$$

Reduce the fraction \frac{-10}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{6}x=-\frac{4}{12}$$

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

$$x^{2}-\frac{5}{6}x=-\frac{1}{3}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=-\frac{1}{3}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{1}{3}+\frac{25}{144}$$

Add -\frac{1}{3} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{23}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{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}.

$$\left(x-\frac{5}{12}\right)^{2}=-\frac{23}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{-\frac{23}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{\sqrt{23}i}{12} x-\frac{5}{12}=-\frac{\sqrt{23}i}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$x=\frac{5+\sqrt{23}i}{12} x=\frac{-\sqrt{23}i+5}{12}$$