Solve 6x^2-17ix-12=0

$$6 x ^ { 2 } - 17 i x - 12 = 0$$

Solve for x

$$x=\frac{3}{2}i=1.5i$$

$$x=\frac{4}{3}i\approx 1.333333333i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}-17ix-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -17i for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{17i±\sqrt{\left(-17i\right)^{2}-4\times 6\left(-12\right)}}{2\times 6}$$

Square -17i.

$$x=\frac{17i±\sqrt{-289-4\times 6\left(-12\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{17i±\sqrt{-289-24\left(-12\right)}}{2\times 6}$$

Multiply -24 times -12.

$$x=\frac{17i±\sqrt{-289+288}}{2\times 6}$$

Add -289 to 288.

$$x=\frac{17i±\sqrt{-1}}{2\times 6}$$

Take the square root of -1.

$$x=\frac{17i±i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{17i±i}{12}$$

Now solve the equation x=\frac{17i±i}{12} when ± is plus. Add 17i to i.

$$x=\frac{18i}{12}$$

Divide 18i by 12.

$$x=\frac{3}{2}i$$

Now solve the equation x=\frac{17i±i}{12} when ± is minus. Subtract i from 17i.

$$x=\frac{16i}{12}$$

Divide 16i by 12.

$$x=\frac{4}{3}i$$

The equation is now solved.

$$x=\frac{3}{2}i x=\frac{4}{3}i$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}-17ix-12=0$$

Add 12 to both sides of the equation.

$$6x^{2}-17ix-12-\left(-12\right)=-\left(-12\right)$$

Subtracting -12 from itself leaves 0.

$$6x^{2}-17ix=-\left(-12\right)$$

Subtract -12 from 0.

$$6x^{2}-17ix=12$$

Divide both sides by 6.

$$\frac{6x^{2}-17ix}{6}=\frac{12}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{-17i}{6}x=\frac{12}{6}$$

Divide -17i by 6.

$$x^{2}-\frac{17}{6}ix=\frac{12}{6}$$

Divide 12 by 6.

$$x^{2}-\frac{17}{6}ix=2$$

Divide -\frac{17}{6}i, the coefficient of the x term, by 2 to get -\frac{17}{12}i. Then add the square of -\frac{17}{12}i to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{6}ix+\left(-\frac{17}{12}i\right)^{2}=2+\left(-\frac{17}{12}i\right)^{2}$$

Square -\frac{17}{12}i.

$$x^{2}-\frac{17}{6}ix-\frac{289}{144}=2-\frac{289}{144}$$

Add 2 to -\frac{289}{144}.

$$x^{2}-\frac{17}{6}ix-\frac{289}{144}=-\frac{1}{144}$$

Factor x^{2}-\frac{17}{6}ix-\frac{289}{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}.

$$\left(x-\frac{17}{12}i\right)^{2}=-\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{12}i\right)^{2}}=\sqrt{-\frac{1}{144}}$$

Simplify.

$$x-\frac{17}{12}i=\frac{1}{12}i x-\frac{17}{12}i=-\frac{1}{12}i$$

Add \frac{17}{12}i to both sides of the equation.

$$x=\frac{3}{2}i x=\frac{4}{3}i$$