Solve 3/2(4x-3)=2(-4x^2-3)

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

Solve for x (complex solution)

$$x=\frac{-3+\sqrt{3}i}{8}\approx -0.375+0.216506351i$$

$$x=\frac{-\sqrt{3}i-3}{8}\approx -0.375-0.216506351i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{3}{2} by 4x-3.

$$6x-\frac{9}{2}=2\left(-4x^{2}-3\right)$$

Use the distributive property to multiply 2 by -4x^{2}-3.

$$6x-\frac{9}{2}=-8x^{2}-6$$

Add 8x^{2} to both sides.

$$6x-\frac{9}{2}+8x^{2}=-6$$

Add 6 to both sides.

$$6x-\frac{9}{2}+8x^{2}+6=0$$

Add -\frac{9}{2} and 6 to get \frac{3}{2}.

$$6x+\frac{3}{2}+8x^{2}=0$$

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.

$$8x^{2}+6x+\frac{3}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 6 for b, and \frac{3}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 8\times \frac{3}{2}}}{2\times 8}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 8\times \frac{3}{2}}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-6±\sqrt{36-32\times \frac{3}{2}}}{2\times 8}$$

Multiply -32 times \frac{3}{2}.

$$x=\frac{-6±\sqrt{36-48}}{2\times 8}$$

Add 36 to -48.

$$x=\frac{-6±\sqrt{-12}}{2\times 8}$$

Take the square root of -12.

$$x=\frac{-6±2\sqrt{3}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-6±2\sqrt{3}i}{16}$$

Now solve the equation x=\frac{-6±2\sqrt{3}i}{16} when ± is plus. Add -6 to 2i\sqrt{3}.

$$x=\frac{-6+2\sqrt{3}i}{16}$$

Divide -6+2i\sqrt{3} by 16.

$$x=\frac{-3+\sqrt{3}i}{8}$$

Now solve the equation x=\frac{-6±2\sqrt{3}i}{16} when ± is minus. Subtract 2i\sqrt{3} from -6.

$$x=\frac{-2\sqrt{3}i-6}{16}$$

Divide -6-2i\sqrt{3} by 16.

$$x=\frac{-\sqrt{3}i-3}{8}$$

The equation is now solved.

$$x=\frac{-3+\sqrt{3}i}{8} x=\frac{-\sqrt{3}i-3}{8}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{3}{2} by 4x-3.

$$6x-\frac{9}{2}=2\left(-4x^{2}-3\right)$$

Use the distributive property to multiply 2 by -4x^{2}-3.

$$6x-\frac{9}{2}=-8x^{2}-6$$

Add 8x^{2} to both sides.

$$6x-\frac{9}{2}+8x^{2}=-6$$

Add \frac{9}{2} to both sides.

$$6x+8x^{2}=-6+\frac{9}{2}$$

Add -6 and \frac{9}{2} to get -\frac{3}{2}.

$$6x+8x^{2}=-\frac{3}{2}$$

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.

$$8x^{2}+6x=-\frac{3}{2}$$

Divide both sides by 8.

$$\frac{8x^{2}+6x}{8}=-\frac{\frac{3}{2}}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{6}{8}x=-\frac{\frac{3}{2}}{8}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{3}{4}x=-\frac{\frac{3}{2}}{8}$$

Divide -\frac{3}{2} by 8.

$$x^{2}+\frac{3}{4}x=-\frac{3}{16}$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{3}{16}+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{3}{16}+\frac{9}{64}$$

Add -\frac{3}{16} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{3}{64}$$

Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{8}\right)^{2}=-\frac{3}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{3}{64}}$$

Simplify.

$$x+\frac{3}{8}=\frac{\sqrt{3}i}{8} x+\frac{3}{8}=-\frac{\sqrt{3}i}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$x=\frac{-3+\sqrt{3}i}{8} x=\frac{-\sqrt{3}i-3}{8}$$