Solve 12x-1/2-5x/1=2x+3/5

$$\frac{ 12x-1 }{ 2 } \frac{ -5x }{ 1 } = \frac{ 2x+3 }{ 5 }$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{751}i}{200}\approx 0.035+0.137021896i$$

$$x=\frac{-\sqrt{751}i+7}{200}\approx 0.035-0.137021896i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10, the least common multiple of 2,5.

$$5\left(12x-1\right)\left(-5\right)x=2\left(2x+3\right)$$

Multiply 5 and -5 to get -25.

$$-25\left(12x-1\right)x=2\left(2x+3\right)$$

Use the distributive property to multiply -25 by 12x-1.

$$\left(-300x+25\right)x=2\left(2x+3\right)$$

Use the distributive property to multiply -300x+25 by x.

$$-300x^{2}+25x=2\left(2x+3\right)$$

Use the distributive property to multiply 2 by 2x+3.

$$-300x^{2}+25x=4x+6$$

Subtract 4x from both sides.

$$-300x^{2}+25x-4x=6$$

Combine 25x and -4x to get 21x.

$$-300x^{2}+21x=6$$

Subtract 6 from both sides.

$$-300x^{2}+21x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -300 for a, 21 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-300\right)\left(-6\right)}}{2\left(-300\right)}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\left(-300\right)\left(-6\right)}}{2\left(-300\right)}$$

Multiply -4 times -300.

$$x=\frac{-21±\sqrt{441+1200\left(-6\right)}}{2\left(-300\right)}$$

Multiply 1200 times -6.

$$x=\frac{-21±\sqrt{441-7200}}{2\left(-300\right)}$$

Add 441 to -7200.

$$x=\frac{-21±\sqrt{-6759}}{2\left(-300\right)}$$

Take the square root of -6759.

$$x=\frac{-21±3\sqrt{751}i}{2\left(-300\right)}$$

Multiply 2 times -300.

$$x=\frac{-21±3\sqrt{751}i}{-600}$$

Now solve the equation x=\frac{-21±3\sqrt{751}i}{-600} when ± is plus. Add -21 to 3i\sqrt{751}.

$$x=\frac{-21+3\sqrt{751}i}{-600}$$

Divide -21+3i\sqrt{751} by -600.

$$x=\frac{-\sqrt{751}i+7}{200}$$

Now solve the equation x=\frac{-21±3\sqrt{751}i}{-600} when ± is minus. Subtract 3i\sqrt{751} from -21.

$$x=\frac{-3\sqrt{751}i-21}{-600}$$

Divide -21-3i\sqrt{751} by -600.

$$x=\frac{7+\sqrt{751}i}{200}$$

The equation is now solved.

$$x=\frac{-\sqrt{751}i+7}{200} x=\frac{7+\sqrt{751}i}{200}$$

Steps for Completing the Square

Multiply both sides of the equation by 10, the least common multiple of 2,5.

$$5\left(12x-1\right)\left(-5\right)x=2\left(2x+3\right)$$

Multiply 5 and -5 to get -25.

$$-25\left(12x-1\right)x=2\left(2x+3\right)$$

Use the distributive property to multiply -25 by 12x-1.

$$\left(-300x+25\right)x=2\left(2x+3\right)$$

Use the distributive property to multiply -300x+25 by x.

$$-300x^{2}+25x=2\left(2x+3\right)$$

Use the distributive property to multiply 2 by 2x+3.

$$-300x^{2}+25x=4x+6$$

Subtract 4x from both sides.

$$-300x^{2}+25x-4x=6$$

Combine 25x and -4x to get 21x.

$$-300x^{2}+21x=6$$

Divide both sides by -300.

$$\frac{-300x^{2}+21x}{-300}=\frac{6}{-300}$$

Dividing by -300 undoes the multiplication by -300.

$$x^{2}+\frac{21}{-300}x=\frac{6}{-300}$$

Reduce the fraction \frac{21}{-300} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{7}{100}x=\frac{6}{-300}$$

Reduce the fraction \frac{6}{-300} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{7}{100}x=-\frac{1}{50}$$

Divide -\frac{7}{100}, the coefficient of the x term, by 2 to get -\frac{7}{200}. Then add the square of -\frac{7}{200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{100}x+\left(-\frac{7}{200}\right)^{2}=-\frac{1}{50}+\left(-\frac{7}{200}\right)^{2}$$

Square -\frac{7}{200} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{100}x+\frac{49}{40000}=-\frac{1}{50}+\frac{49}{40000}$$

Add -\frac{1}{50} to \frac{49}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{100}x+\frac{49}{40000}=-\frac{751}{40000}$$

Factor x^{2}-\frac{7}{100}x+\frac{49}{40000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{200}\right)^{2}=-\frac{751}{40000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{200}\right)^{2}}=\sqrt{-\frac{751}{40000}}$$

Simplify.

$$x-\frac{7}{200}=\frac{\sqrt{751}i}{200} x-\frac{7}{200}=-\frac{\sqrt{751}i}{200}$$

Add \frac{7}{200} to both sides of the equation.

$$x=\frac{7+\sqrt{751}i}{200} x=\frac{-\sqrt{751}i+7}{200}$$