Solve x(x+3)/3=5x-1/4

$$\frac { x ( x + 3 ) } { 3 } = \frac { 5 x - 1 } { 4 }$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{39}i}{8}\approx 0.375+0.78062475i$$

$$x=\frac{-\sqrt{39}i+3}{8}\approx 0.375-0.78062475i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4x\left(x+3\right)=3\left(5x-1\right)$$

Use the distributive property to multiply 4x by x+3.

$$4x^{2}+12x=3\left(5x-1\right)$$

Use the distributive property to multiply 3 by 5x-1.

$$4x^{2}+12x=15x-3$$

Subtract 15x from both sides.

$$4x^{2}+12x-15x=-3$$

Combine 12x and -15x to get -3x.

$$4x^{2}-3x=-3$$

Add 3 to both sides.

$$4x^{2}-3x+3=0$$

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

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 4\times 3}}{2\times 4}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-3\right)±\sqrt{9-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$x=\frac{-\left(-3\right)±\sqrt{9-48}}{2\times 4}$$

Add 9 to -48.

$$x=\frac{-\left(-3\right)±\sqrt{-39}}{2\times 4}$$

Take the square root of -39.

$$x=\frac{-\left(-3\right)±\sqrt{39}i}{2\times 4}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{39}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{3±\sqrt{39}i}{8}$$

Now solve the equation x=\frac{3±\sqrt{39}i}{8} when ± is plus. Add 3 to i\sqrt{39}.

$$x=\frac{3+\sqrt{39}i}{8}$$

Now solve the equation x=\frac{3±\sqrt{39}i}{8} when ± is minus. Subtract i\sqrt{39} from 3.

$$x=\frac{-\sqrt{39}i+3}{8}$$

The equation is now solved.

$$x=\frac{3+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+3}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4x\left(x+3\right)=3\left(5x-1\right)$$

Use the distributive property to multiply 4x by x+3.

$$4x^{2}+12x=3\left(5x-1\right)$$

Use the distributive property to multiply 3 by 5x-1.

$$4x^{2}+12x=15x-3$$

Subtract 15x from both sides.

$$4x^{2}+12x-15x=-3$$

Combine 12x and -15x to get -3x.

$$4x^{2}-3x=-3$$

Divide both sides by 4.

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

Dividing by 4 undoes the multiplication by 4.

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

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.

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

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

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

Add -\frac{3}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{39}{64}$$

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}.

$$\left(x-\frac{3}{8}\right)^{2}=-\frac{39}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{-\frac{39}{64}}$$

Simplify.

$$x-\frac{3}{8}=\frac{\sqrt{39}i}{8} x-\frac{3}{8}=-\frac{\sqrt{39}i}{8}$$

Add \frac{3}{8} to both sides of the equation.

$$x=\frac{3+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+3}{8}$$