Solve 1-2x/6x^2+3x+2x+1/14x^2-7x=1/12x^2-3

$$\frac { 1 - 2 x } { 6 x ^ { 2 } + 3 x } + \frac { 2 x + 1 } { 14 x ^ { 2 } - 7 x } = \frac { 1 } { 12 x ^ { 2 } - 3 }$$

Solve for x

$$x = \frac{7 \sqrt{17} + 33}{32} \approx 1.933179356$$

$$x=\frac{33-7\sqrt{17}}{32}\approx 0.129320644$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{1}{2},0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 21x\left(2x-1\right)\left(2x+1\right), the least common multiple of 6x^{2}+3x,14x^{2}-7x,12x^{2}-3.

$$\left(14x-7\right)\left(1-2x\right)+\left(6x+3\right)\left(2x+1\right)=7x$$

Use the distributive property to multiply 14x-7 by 1-2x and combine like terms.

$$28x-28x^{2}-7+\left(6x+3\right)\left(2x+1\right)=7x$$

Use the distributive property to multiply 6x+3 by 2x+1 and combine like terms.

$$28x-28x^{2}-7+12x^{2}+12x+3=7x$$

Combine -28x^{2} and 12x^{2} to get -16x^{2}.

$$28x-16x^{2}-7+12x+3=7x$$

Combine 28x and 12x to get 40x.

$$40x-16x^{2}-7+3=7x$$

Add -7 and 3 to get -4.

$$40x-16x^{2}-4=7x$$

Subtract 7x from both sides.

$$40x-16x^{2}-4-7x=0$$

Combine 40x and -7x to get 33x.

$$33x-16x^{2}-4=0$$

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.

$$-16x^{2}+33x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 33 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-16\right)\left(-4\right)}}{2\left(-16\right)}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\left(-16\right)\left(-4\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-33±\sqrt{1089+64\left(-4\right)}}{2\left(-16\right)}$$

Multiply 64 times -4.

$$x=\frac{-33±\sqrt{1089-256}}{2\left(-16\right)}$$

Add 1089 to -256.

$$x=\frac{-33±\sqrt{833}}{2\left(-16\right)}$$

Take the square root of 833.

$$x=\frac{-33±7\sqrt{17}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-33±7\sqrt{17}}{-32}$$

Now solve the equation x=\frac{-33±7\sqrt{17}}{-32} when ± is plus. Add -33 to 7\sqrt{17}.

$$x=\frac{7\sqrt{17}-33}{-32}$$

Divide -33+7\sqrt{17} by -32.

$$x=\frac{33-7\sqrt{17}}{32}$$

Now solve the equation x=\frac{-33±7\sqrt{17}}{-32} when ± is minus. Subtract 7\sqrt{17} from -33.

$$x=\frac{-7\sqrt{17}-33}{-32}$$

Divide -33-7\sqrt{17} by -32.

$$x=\frac{7\sqrt{17}+33}{32}$$

The equation is now solved.

$$x=\frac{33-7\sqrt{17}}{32} x=\frac{7\sqrt{17}+33}{32}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{1}{2},0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 21x\left(2x-1\right)\left(2x+1\right), the least common multiple of 6x^{2}+3x,14x^{2}-7x,12x^{2}-3.

$$\left(14x-7\right)\left(1-2x\right)+\left(6x+3\right)\left(2x+1\right)=7x$$

Use the distributive property to multiply 14x-7 by 1-2x and combine like terms.

$$28x-28x^{2}-7+\left(6x+3\right)\left(2x+1\right)=7x$$

Use the distributive property to multiply 6x+3 by 2x+1 and combine like terms.

$$28x-28x^{2}-7+12x^{2}+12x+3=7x$$

Combine -28x^{2} and 12x^{2} to get -16x^{2}.

$$28x-16x^{2}-7+12x+3=7x$$

Combine 28x and 12x to get 40x.

$$40x-16x^{2}-7+3=7x$$

Add -7 and 3 to get -4.

$$40x-16x^{2}-4=7x$$

Subtract 7x from both sides.

$$40x-16x^{2}-4-7x=0$$

Combine 40x and -7x to get 33x.

$$33x-16x^{2}-4=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$33x-16x^{2}=4$$

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.

$$-16x^{2}+33x=4$$

Divide both sides by -16.

$$\frac{-16x^{2}+33x}{-16}=\frac{4}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{33}{-16}x=\frac{4}{-16}$$

Divide 33 by -16.

$$x^{2}-\frac{33}{16}x=\frac{4}{-16}$$

Reduce the fraction \frac{4}{-16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{33}{16}x=-\frac{1}{4}$$

Divide -\frac{33}{16}, the coefficient of the x term, by 2 to get -\frac{33}{32}. Then add the square of -\frac{33}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{33}{16}x+\left(-\frac{33}{32}\right)^{2}=-\frac{1}{4}+\left(-\frac{33}{32}\right)^{2}$$

Square -\frac{33}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{33}{16}x+\frac{1089}{1024}=-\frac{1}{4}+\frac{1089}{1024}$$

Add -\frac{1}{4} to \frac{1089}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{33}{16}x+\frac{1089}{1024}=\frac{833}{1024}$$

Factor x^{2}-\frac{33}{16}x+\frac{1089}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{33}{32}\right)^{2}=\frac{833}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{32}\right)^{2}}=\sqrt{\frac{833}{1024}}$$

Simplify.

$$x-\frac{33}{32}=\frac{7\sqrt{17}}{32} x-\frac{33}{32}=-\frac{7\sqrt{17}}{32}$$

Add \frac{33}{32} to both sides of the equation.

$$x=\frac{7\sqrt{17}+33}{32} x=\frac{33-7\sqrt{17}}{32}$$