Solve 9-x^2/9-16x^2+3/3-4x=x+12/6+8x

$$\frac { 9 - x ^ { 2 } } { 9 - 16 x ^ { 2 } } + \frac { 3 } { 3 - 4 x } = \frac { x + 12 } { 6 + 8 x }$$

Solve for x

$$x = -\frac{69}{2} = -34\frac{1}{2} = -34.5$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to any of the values -\frac{3}{4},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 2\left(4x-3\right)\left(4x+3\right), the least common multiple of 9-16x^{2},3-4x,6+8x.

$$-2\left(9-x^{2}\right)+\left(-6-8x\right)\times 3=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply -2 by 9-x^{2}.

$$-18+2x^{2}+\left(-6-8x\right)\times 3=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply -6-8x by 3.

$$-18+2x^{2}-18-24x=\left(4x-3\right)\left(x+12\right)$$

Subtract 18 from -18 to get -36.

$$-36+2x^{2}-24x=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply 4x-3 by x+12 and combine like terms.

$$-36+2x^{2}-24x=4x^{2}+45x-36$$

Subtract 4x^{2} from both sides.

$$-36+2x^{2}-24x-4x^{2}=45x-36$$

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

$$-36-2x^{2}-24x=45x-36$$

Subtract 45x from both sides.

$$-36-2x^{2}-24x-45x=-36$$

Combine -24x and -45x to get -69x.

$$-36-2x^{2}-69x=-36$$

Add 36 to both sides.

$$-36-2x^{2}-69x+36=0$$

Add -36 and 36 to get 0.

$$-2x^{2}-69x=0$$

Factor out x.

$$x\left(-2x-69\right)=0$$

To find equation solutions, solve x=0 and -2x-69=0.

$$x=0 x=-\frac{69}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{3}{4},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 2\left(4x-3\right)\left(4x+3\right), the least common multiple of 9-16x^{2},3-4x,6+8x.

$$-2\left(9-x^{2}\right)+\left(-6-8x\right)\times 3=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply -2 by 9-x^{2}.

$$-18+2x^{2}+\left(-6-8x\right)\times 3=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply -6-8x by 3.

$$-18+2x^{2}-18-24x=\left(4x-3\right)\left(x+12\right)$$

Subtract 18 from -18 to get -36.

$$-36+2x^{2}-24x=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply 4x-3 by x+12 and combine like terms.

$$-36+2x^{2}-24x=4x^{2}+45x-36$$

Subtract 4x^{2} from both sides.

$$-36+2x^{2}-24x-4x^{2}=45x-36$$

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

$$-36-2x^{2}-24x=45x-36$$

Subtract 45x from both sides.

$$-36-2x^{2}-24x-45x=-36$$

Combine -24x and -45x to get -69x.

$$-36-2x^{2}-69x=-36$$

Add 36 to both sides.

$$-36-2x^{2}-69x+36=0$$

Add -36 and 36 to get 0.

$$-2x^{2}-69x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -69 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-69\right)±\sqrt{\left(-69\right)^{2}}}{2\left(-2\right)}$$

Take the square root of \left(-69\right)^{2}.

$$x=\frac{-\left(-69\right)±69}{2\left(-2\right)}$$

The opposite of -69 is 69.

$$x=\frac{69±69}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{69±69}{-4}$$

Now solve the equation x=\frac{69±69}{-4} when ± is plus. Add 69 to 69.

$$x=\frac{138}{-4}$$

Reduce the fraction \frac{138}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{69}{2}$$

Now solve the equation x=\frac{69±69}{-4} when ± is minus. Subtract 69 from 69.

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

The equation is now solved.

$$x=-\frac{69}{2} x=0$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{3}{4},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 2\left(4x-3\right)\left(4x+3\right), the least common multiple of 9-16x^{2},3-4x,6+8x.

$$-2\left(9-x^{2}\right)+\left(-6-8x\right)\times 3=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply -2 by 9-x^{2}.

$$-18+2x^{2}+\left(-6-8x\right)\times 3=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply -6-8x by 3.

$$-18+2x^{2}-18-24x=\left(4x-3\right)\left(x+12\right)$$

Subtract 18 from -18 to get -36.

$$-36+2x^{2}-24x=\left(4x-3\right)\left(x+12\right)$$

Use the distributive property to multiply 4x-3 by x+12 and combine like terms.

$$-36+2x^{2}-24x=4x^{2}+45x-36$$

Subtract 4x^{2} from both sides.

$$-36+2x^{2}-24x-4x^{2}=45x-36$$

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

$$-36-2x^{2}-24x=45x-36$$

Subtract 45x from both sides.

$$-36-2x^{2}-24x-45x=-36$$

Combine -24x and -45x to get -69x.

$$-36-2x^{2}-69x=-36$$

Add 36 to both sides.

$$-2x^{2}-69x=-36+36$$

Add -36 and 36 to get 0.

$$-2x^{2}-69x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}-69x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{69}{-2}\right)x=\frac{0}{-2}$$

Divide -69 by -2.

$$x^{2}+\frac{69}{2}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}+\frac{69}{2}x=0$$

Divide \frac{69}{2}, the coefficient of the x term, by 2 to get \frac{69}{4}. Then add the square of \frac{69}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{69}{2}x+\left(\frac{69}{4}\right)^{2}=\left(\frac{69}{4}\right)^{2}$$

Square \frac{69}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{69}{2}x+\frac{4761}{16}=\frac{4761}{16}$$

Factor x^{2}+\frac{69}{2}x+\frac{4761}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{69}{4}\right)^{2}=\frac{4761}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{69}{4}\right)^{2}}=\sqrt{\frac{4761}{16}}$$

Simplify.

$$x+\frac{69}{4}=\frac{69}{4} x+\frac{69}{4}=-\frac{69}{4}$$

Subtract \frac{69}{4} from both sides of the equation.

$$x=0 x=-\frac{69}{2}$$