Solve 3(4x^2+3x)=-17

$$3 ( 4 x ^ { 2 } + 3 x ) = - 17$$

Solve for x (complex solution)

$$x=\frac{7\sqrt{15}i}{24}-\frac{3}{8}\approx -0.375+1.129620143i$$

$$x=-\frac{7\sqrt{15}i}{24}-\frac{3}{8}\approx -0.375-1.129620143i$$

Steps Using the Quadratic Formula

Divide both sides by 3.

$$4x^{2}+3x=-\frac{17}{3}$$

Add \frac{17}{3} to both sides.

$$4x^{2}+3x+\frac{17}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 3 for b, and \frac{17}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 4\times \frac{17}{3}}}{2\times 4}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 4\times \frac{17}{3}}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-3±\sqrt{9-16\times \frac{17}{3}}}{2\times 4}$$

Multiply -16 times \frac{17}{3}.

$$x=\frac{-3±\sqrt{9-\frac{272}{3}}}{2\times 4}$$

Add 9 to -\frac{272}{3}.

$$x=\frac{-3±\sqrt{-\frac{245}{3}}}{2\times 4}$$

Take the square root of -\frac{245}{3}.

$$x=\frac{-3±\frac{7\sqrt{15}i}{3}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-3±\frac{7\sqrt{15}i}{3}}{8}$$

Now solve the equation x=\frac{-3±\frac{7\sqrt{15}i}{3}}{8} when ± is plus. Add -3 to \frac{7i\sqrt{15}}{3}.

$$x=\frac{\frac{7\sqrt{15}i}{3}-3}{8}$$

Divide -3+\frac{7i\sqrt{15}}{3} by 8.

$$x=\frac{7\sqrt{15}i}{24}-\frac{3}{8}$$

Now solve the equation x=\frac{-3±\frac{7\sqrt{15}i}{3}}{8} when ± is minus. Subtract \frac{7i\sqrt{15}}{3} from -3.

$$x=\frac{-\frac{7\sqrt{15}i}{3}-3}{8}$$

Divide -3-\frac{7i\sqrt{15}}{3} by 8.

$$x=-\frac{7\sqrt{15}i}{24}-\frac{3}{8}$$

The equation is now solved.

$$x=\frac{7\sqrt{15}i}{24}-\frac{3}{8} x=-\frac{7\sqrt{15}i}{24}-\frac{3}{8}$$

Steps for Completing the Square

Divide both sides by 3.

$$4x^{2}+3x=-\frac{17}{3}$$

Divide both sides by 4.

$$\frac{4x^{2}+3x}{4}=-\frac{\frac{17}{3}}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{3}{4}x=-\frac{\frac{17}{3}}{4}$$

Divide -\frac{17}{3} by 4.

$$x^{2}+\frac{3}{4}x=-\frac{17}{12}$$

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.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{17}{12}+\left(\frac{3}{8}\right)^{2}$$

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

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{17}{12}+\frac{9}{64}$$

Add -\frac{17}{12} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{245}{192}$$

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}.

$$\left(x+\frac{3}{8}\right)^{2}=-\frac{245}{192}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{245}{192}}$$

Simplify.

$$x+\frac{3}{8}=\frac{7\sqrt{15}i}{24} x+\frac{3}{8}=-\frac{7\sqrt{15}i}{24}$$

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

$$x=\frac{7\sqrt{15}i}{24}-\frac{3}{8} x=-\frac{7\sqrt{15}i}{24}-\frac{3}{8}$$