Solve 3left(2xright)^2+5x+30=0

$$3 { \left(2x \right) }^{ 2 } +5x+30 = 0$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{1415}i}{24}\approx -0.208333333+1.567353573i$$

$$x=\frac{-\sqrt{1415}i-5}{24}\approx -0.208333333-1.567353573i$$

Steps Using the Quadratic Formula

Expand \left(2x\right)^{2}.

$$3\times 2^{2}x^{2}+5x+30=0$$

Calculate 2 to the power of 2 and get 4.

$$3\times 4x^{2}+5x+30=0$$

Multiply 3 and 4 to get 12.

$$12x^{2}+5x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 5 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 12\times 30}}{2\times 12}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 12\times 30}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-5±\sqrt{25-48\times 30}}{2\times 12}$$

Multiply -48 times 30.

$$x=\frac{-5±\sqrt{25-1440}}{2\times 12}$$

Add 25 to -1440.

$$x=\frac{-5±\sqrt{-1415}}{2\times 12}$$

Take the square root of -1415.

$$x=\frac{-5±\sqrt{1415}i}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-5±\sqrt{1415}i}{24}$$

Now solve the equation x=\frac{-5±\sqrt{1415}i}{24} when ± is plus. Add -5 to i\sqrt{1415}.

$$x=\frac{-5+\sqrt{1415}i}{24}$$

Now solve the equation x=\frac{-5±\sqrt{1415}i}{24} when ± is minus. Subtract i\sqrt{1415} from -5.

$$x=\frac{-\sqrt{1415}i-5}{24}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{1415}i}{24} x=\frac{-\sqrt{1415}i-5}{24}$$

Steps for Completing the Square

Expand \left(2x\right)^{2}.

$$3\times 2^{2}x^{2}+5x+30=0$$

Calculate 2 to the power of 2 and get 4.

$$3\times 4x^{2}+5x+30=0$$

Multiply 3 and 4 to get 12.

$$12x^{2}+5x+30=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$12x^{2}+5x=-30$$

Divide both sides by 12.

$$\frac{12x^{2}+5x}{12}=-\frac{30}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{5}{12}x=-\frac{30}{12}$$

Reduce the fraction \frac{-30}{12} to lowest terms by extracting and canceling out 6.

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

Divide \frac{5}{12}, the coefficient of the x term, by 2 to get \frac{5}{24}. Then add the square of \frac{5}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{12}x+\left(\frac{5}{24}\right)^{2}=-\frac{5}{2}+\left(\frac{5}{24}\right)^{2}$$

Square \frac{5}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{12}x+\frac{25}{576}=-\frac{5}{2}+\frac{25}{576}$$

Add -\frac{5}{2} to \frac{25}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{12}x+\frac{25}{576}=-\frac{1415}{576}$$

Factor x^{2}+\frac{5}{12}x+\frac{25}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{24}\right)^{2}=-\frac{1415}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{24}\right)^{2}}=\sqrt{-\frac{1415}{576}}$$

Simplify.

$$x+\frac{5}{24}=\frac{\sqrt{1415}i}{24} x+\frac{5}{24}=-\frac{\sqrt{1415}i}{24}$$

Subtract \frac{5}{24} from both sides of the equation.

$$x=\frac{-5+\sqrt{1415}i}{24} x=\frac{-\sqrt{1415}i-5}{24}$$