Solve 2x/1-x^2=-12/5

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

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-1\right)\left(x+1\right), the least common multiple of 1-x^{2},5.

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

Multiply -5 and 2 to get -10.

$$-10x=-12\left(x-1\right)\left(x+1\right)$$

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

$$-10x=\left(-12x+12\right)\left(x+1\right)$$

Use the distributive property to multiply -12x+12 by x+1 and combine like terms.

$$-10x=-12x^{2}+12$$

Add 12x^{2} to both sides.

$$-10x+12x^{2}=12$$

Subtract 12 from both sides.

$$-10x+12x^{2}-12=0$$

Divide both sides by 2.

$$-5x+6x^{2}-6=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$6x^{2}-5x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-6\right)=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -5.

$$a=-9 b=4$$

Rewrite 6x^{2}-5x-6 as \left(6x^{2}-9x\right)+\left(4x-6\right).

$$\left(6x^{2}-9x\right)+\left(4x-6\right)$$

Factor out 3x in the first and 2 in the second group.

$$3x\left(2x-3\right)+2\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(3x+2\right)$$

To find equation solutions, solve 2x-3=0 and 3x+2=0.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-1\right)\left(x+1\right), the least common multiple of 1-x^{2},5.

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

Multiply -5 and 2 to get -10.

$$-10x=-12\left(x-1\right)\left(x+1\right)$$

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

$$-10x=\left(-12x+12\right)\left(x+1\right)$$

Use the distributive property to multiply -12x+12 by x+1 and combine like terms.

$$-10x=-12x^{2}+12$$

Add 12x^{2} to both sides.

$$-10x+12x^{2}=12$$

Subtract 12 from both sides.

$$-10x+12x^{2}-12=0$$

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.

$$12x^{2}-10x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -10 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 12\left(-12\right)}}{2\times 12}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 12\left(-12\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-10\right)±\sqrt{100-48\left(-12\right)}}{2\times 12}$$

Multiply -48 times -12.

$$x=\frac{-\left(-10\right)±\sqrt{100+576}}{2\times 12}$$

Add 100 to 576.

$$x=\frac{-\left(-10\right)±\sqrt{676}}{2\times 12}$$

Take the square root of 676.

$$x=\frac{-\left(-10\right)±26}{2\times 12}$$

The opposite of -10 is 10.

$$x=\frac{10±26}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{10±26}{24}$$

Now solve the equation x=\frac{10±26}{24} when ± is plus. Add 10 to 26.

$$x=\frac{36}{24}$$

Reduce the fraction \frac{36}{24} to lowest terms by extracting and canceling out 12.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{10±26}{24} when ± is minus. Subtract 26 from 10.

$$x=-\frac{16}{24}$$

Reduce the fraction \frac{-16}{24} to lowest terms by extracting and canceling out 8.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-1\right)\left(x+1\right), the least common multiple of 1-x^{2},5.

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

Multiply -5 and 2 to get -10.

$$-10x=-12\left(x-1\right)\left(x+1\right)$$

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

$$-10x=\left(-12x+12\right)\left(x+1\right)$$

Use the distributive property to multiply -12x+12 by x+1 and combine like terms.

$$-10x=-12x^{2}+12$$

Add 12x^{2} to both sides.

$$-10x+12x^{2}=12$$

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.

$$12x^{2}-10x=12$$

Divide both sides by 12.

$$\frac{12x^{2}-10x}{12}=\frac{12}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{10}{12}\right)x=\frac{12}{12}$$

Reduce the fraction \frac{-10}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{6}x=\frac{12}{12}$$

Divide 12 by 12.

$$x^{2}-\frac{5}{6}x=1$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=1+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=1+\frac{25}{144}$$

Add 1 to \frac{25}{144}.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{169}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{169}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{13}{12} x-\frac{5}{12}=-\frac{13}{12}$$

Add \frac{5}{12} to both sides of the equation.

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