Solve -7x-3/2x-3x^2-4x-1/9x^2-4=-2x+1/2x+3x^2

$$- \frac { 7 x - 3 } { 2 x - 3 x ^ { 2 } } - \frac { 4 x - 1 } { 9 x ^ { 2 } - 4 } = - \frac { 2 x + 1 } { 2 x + 3 x ^ { 2 } }$$

Solve for x

$$x=\frac{-\sqrt{761}-5}{46}\approx -0.708396271$$

$$x=\frac{\sqrt{761}-5}{46}\approx 0.491004966$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{3},0,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by x\left(3x-2\right)\left(3x+2\right), the least common multiple of 2x-3x^{2},9x^{2}-4,2x+3x^{2}.

$$-\left(-2-3x\right)\left(7x-3\right)-x\left(4x-1\right)=-\left(3x-2\right)\left(2x+1\right)$$

Use the distributive property to multiply -2-3x by 7x-3 and combine like terms.

$$-\left(-5x+6-21x^{2}\right)-x\left(4x-1\right)=-\left(3x-2\right)\left(2x+1\right)$$

To find the opposite of -5x+6-21x^{2}, find the opposite of each term.

$$5x-6+21x^{2}-x\left(4x-1\right)=-\left(3x-2\right)\left(2x+1\right)$$

Use the distributive property to multiply x by 4x-1.

$$5x-6+21x^{2}-\left(4x^{2}-x\right)=-\left(3x-2\right)\left(2x+1\right)$$

To find the opposite of 4x^{2}-x, find the opposite of each term.

$$5x-6+21x^{2}-4x^{2}+x=-\left(3x-2\right)\left(2x+1\right)$$

Combine 21x^{2} and -4x^{2} to get 17x^{2}.

$$5x-6+17x^{2}+x=-\left(3x-2\right)\left(2x+1\right)$$

Combine 5x and x to get 6x.

$$6x-6+17x^{2}=-\left(3x-2\right)\left(2x+1\right)$$

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

$$6x-6+17x^{2}=-\left(6x^{2}-x-2\right)$$

To find the opposite of 6x^{2}-x-2, find the opposite of each term.

$$6x-6+17x^{2}=-6x^{2}+x+2$$

Add 6x^{2} to both sides.

$$6x-6+17x^{2}+6x^{2}=x+2$$

Combine 17x^{2} and 6x^{2} to get 23x^{2}.

$$6x-6+23x^{2}=x+2$$

Subtract x from both sides.

$$6x-6+23x^{2}-x=2$$

Combine 6x and -x to get 5x.

$$5x-6+23x^{2}=2$$

Subtract 2 from both sides.

$$5x-6+23x^{2}-2=0$$

Subtract 2 from -6 to get -8.

$$5x-8+23x^{2}=0$$

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.

$$23x^{2}+5x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 23 for a, 5 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 23\left(-8\right)}}{2\times 23}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 23\left(-8\right)}}{2\times 23}$$

Multiply -4 times 23.

$$x=\frac{-5±\sqrt{25-92\left(-8\right)}}{2\times 23}$$

Multiply -92 times -8.

$$x=\frac{-5±\sqrt{25+736}}{2\times 23}$$

Add 25 to 736.

$$x=\frac{-5±\sqrt{761}}{2\times 23}$$

Multiply 2 times 23.

$$x=\frac{-5±\sqrt{761}}{46}$$

Now solve the equation x=\frac{-5±\sqrt{761}}{46} when ± is plus. Add -5 to \sqrt{761}.

$$x=\frac{\sqrt{761}-5}{46}$$

Now solve the equation x=\frac{-5±\sqrt{761}}{46} when ± is minus. Subtract \sqrt{761} from -5.

$$x=\frac{-\sqrt{761}-5}{46}$$

The equation is now solved.

$$x=\frac{\sqrt{761}-5}{46} x=\frac{-\sqrt{761}-5}{46}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{3},0,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by x\left(3x-2\right)\left(3x+2\right), the least common multiple of 2x-3x^{2},9x^{2}-4,2x+3x^{2}.

$$-\left(-2-3x\right)\left(7x-3\right)-x\left(4x-1\right)=-\left(3x-2\right)\left(2x+1\right)$$

Use the distributive property to multiply -2-3x by 7x-3 and combine like terms.

$$-\left(-5x+6-21x^{2}\right)-x\left(4x-1\right)=-\left(3x-2\right)\left(2x+1\right)$$

To find the opposite of -5x+6-21x^{2}, find the opposite of each term.

$$5x-6+21x^{2}-x\left(4x-1\right)=-\left(3x-2\right)\left(2x+1\right)$$

Use the distributive property to multiply x by 4x-1.

$$5x-6+21x^{2}-\left(4x^{2}-x\right)=-\left(3x-2\right)\left(2x+1\right)$$

To find the opposite of 4x^{2}-x, find the opposite of each term.

$$5x-6+21x^{2}-4x^{2}+x=-\left(3x-2\right)\left(2x+1\right)$$

Combine 21x^{2} and -4x^{2} to get 17x^{2}.

$$5x-6+17x^{2}+x=-\left(3x-2\right)\left(2x+1\right)$$

Combine 5x and x to get 6x.

$$6x-6+17x^{2}=-\left(3x-2\right)\left(2x+1\right)$$

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

$$6x-6+17x^{2}=-\left(6x^{2}-x-2\right)$$

To find the opposite of 6x^{2}-x-2, find the opposite of each term.

$$6x-6+17x^{2}=-6x^{2}+x+2$$

Add 6x^{2} to both sides.

$$6x-6+17x^{2}+6x^{2}=x+2$$

Combine 17x^{2} and 6x^{2} to get 23x^{2}.

$$6x-6+23x^{2}=x+2$$

Subtract x from both sides.

$$6x-6+23x^{2}-x=2$$

Combine 6x and -x to get 5x.

$$5x-6+23x^{2}=2$$

Add 6 to both sides.

$$5x+23x^{2}=2+6$$

Add 2 and 6 to get 8.

$$5x+23x^{2}=8$$

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.

$$23x^{2}+5x=8$$

Divide both sides by 23.

$$\frac{23x^{2}+5x}{23}=\frac{8}{23}$$

Dividing by 23 undoes the multiplication by 23.

$$x^{2}+\frac{5}{23}x=\frac{8}{23}$$

Divide \frac{5}{23}, the coefficient of the x term, by 2 to get \frac{5}{46}. Then add the square of \frac{5}{46} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{23}x+\left(\frac{5}{46}\right)^{2}=\frac{8}{23}+\left(\frac{5}{46}\right)^{2}$$

Square \frac{5}{46} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{23}x+\frac{25}{2116}=\frac{8}{23}+\frac{25}{2116}$$

Add \frac{8}{23} to \frac{25}{2116} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{23}x+\frac{25}{2116}=\frac{761}{2116}$$

Factor x^{2}+\frac{5}{23}x+\frac{25}{2116}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{46}\right)^{2}=\frac{761}{2116}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{46}\right)^{2}}=\sqrt{\frac{761}{2116}}$$

Simplify.

$$x+\frac{5}{46}=\frac{\sqrt{761}}{46} x+\frac{5}{46}=-\frac{\sqrt{761}}{46}$$

Subtract \frac{5}{46} from both sides of the equation.

$$x=\frac{\sqrt{761}-5}{46} x=\frac{-\sqrt{761}-5}{46}$$