Solve 64/6x^2-x-2+7/2x^2+9x+4=10x/3x^2+10x-8

$$\frac { 64 } { 6 x ^ { 2 } - x - 2 } + \frac { 7 } { 2 x ^ { 2 } + 9 x + 4 } = \frac { 10 x } { 3 x ^ { 2 } + 10 x - 8 }$$

Solve for x

$$x=\frac{\sqrt{24985}}{40}+\frac{15}{8}\approx 5.826661043$$

$$x=-\frac{\sqrt{24985}}{40}+\frac{15}{8}\approx -2.076661043$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-\frac{1}{2},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(3x-2\right)\left(\frac{1}{2}x+2\right)\left(2x+1\right), the least common multiple of 6x^{2}-x-2,2x^{2}+9x+4,3x^{2}+10x-8.

$$\left(4+x\right)\times 64+\left(3x-2\right)\times 7=\left(2x+1\right)\times 10x$$

Use the distributive property to multiply 4+x by 64.

$$256+64x+\left(3x-2\right)\times 7=\left(2x+1\right)\times 10x$$

Use the distributive property to multiply 3x-2 by 7.

$$256+64x+21x-14=\left(2x+1\right)\times 10x$$

Combine 64x and 21x to get 85x.

$$256+85x-14=\left(2x+1\right)\times 10x$$

Subtract 14 from 256 to get 242.

$$242+85x=\left(2x+1\right)\times 10x$$

Use the distributive property to multiply 2x+1 by 10.

$$242+85x=\left(20x+10\right)x$$

Use the distributive property to multiply 20x+10 by x.

$$242+85x=20x^{2}+10x$$

Subtract 20x^{2} from both sides.

$$242+85x-20x^{2}=10x$$

Subtract 10x from both sides.

$$242+85x-20x^{2}-10x=0$$

Combine 85x and -10x to get 75x.

$$242+75x-20x^{2}=0$$

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.

$$-20x^{2}+75x+242=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 75 for b, and 242 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-75±\sqrt{75^{2}-4\left(-20\right)\times 242}}{2\left(-20\right)}$$

Square 75.

$$x=\frac{-75±\sqrt{5625-4\left(-20\right)\times 242}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-75±\sqrt{5625+80\times 242}}{2\left(-20\right)}$$

Multiply 80 times 242.

$$x=\frac{-75±\sqrt{5625+19360}}{2\left(-20\right)}$$

Add 5625 to 19360.

$$x=\frac{-75±\sqrt{24985}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-75±\sqrt{24985}}{-40}$$

Now solve the equation x=\frac{-75±\sqrt{24985}}{-40} when ± is plus. Add -75 to \sqrt{24985}.

$$x=\frac{\sqrt{24985}-75}{-40}$$

Divide -75+\sqrt{24985} by -40.

$$x=-\frac{\sqrt{24985}}{40}+\frac{15}{8}$$

Now solve the equation x=\frac{-75±\sqrt{24985}}{-40} when ± is minus. Subtract \sqrt{24985} from -75.

$$x=\frac{-\sqrt{24985}-75}{-40}$$

Divide -75-\sqrt{24985} by -40.

$$x=\frac{\sqrt{24985}}{40}+\frac{15}{8}$$

The equation is now solved.

$$x=-\frac{\sqrt{24985}}{40}+\frac{15}{8} x=\frac{\sqrt{24985}}{40}+\frac{15}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-\frac{1}{2},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(3x-2\right)\left(\frac{1}{2}x+2\right)\left(2x+1\right), the least common multiple of 6x^{2}-x-2,2x^{2}+9x+4,3x^{2}+10x-8.

$$\left(4+x\right)\times 64+\left(3x-2\right)\times 7=\left(2x+1\right)\times 10x$$

Use the distributive property to multiply 4+x by 64.

$$256+64x+\left(3x-2\right)\times 7=\left(2x+1\right)\times 10x$$

Use the distributive property to multiply 3x-2 by 7.

$$256+64x+21x-14=\left(2x+1\right)\times 10x$$

Combine 64x and 21x to get 85x.

$$256+85x-14=\left(2x+1\right)\times 10x$$

Subtract 14 from 256 to get 242.

$$242+85x=\left(2x+1\right)\times 10x$$

Use the distributive property to multiply 2x+1 by 10.

$$242+85x=\left(20x+10\right)x$$

Use the distributive property to multiply 20x+10 by x.

$$242+85x=20x^{2}+10x$$

Subtract 20x^{2} from both sides.

$$242+85x-20x^{2}=10x$$

Subtract 10x from both sides.

$$242+85x-20x^{2}-10x=0$$

Combine 85x and -10x to get 75x.

$$242+75x-20x^{2}=0$$

Subtract 242 from both sides. Anything subtracted from zero gives its negation.

$$75x-20x^{2}=-242$$

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.

$$-20x^{2}+75x=-242$$

Divide both sides by -20.

$$\frac{-20x^{2}+75x}{-20}=-\frac{242}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{75}{-20}x=-\frac{242}{-20}$$

Reduce the fraction \frac{75}{-20} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{15}{4}x=-\frac{242}{-20}$$

Reduce the fraction \frac{-242}{-20} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{4}x=\frac{121}{10}$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=\frac{121}{10}+\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{121}{10}+\frac{225}{64}$$

Add \frac{121}{10} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{4997}{320}$$

Factor x^{2}-\frac{15}{4}x+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{4997}{320}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{4997}{320}}$$

Simplify.

$$x-\frac{15}{8}=\frac{\sqrt{24985}}{40} x-\frac{15}{8}=-\frac{\sqrt{24985}}{40}$$

Add \frac{15}{8} to both sides of the equation.

$$x=\frac{\sqrt{24985}}{40}+\frac{15}{8} x=-\frac{\sqrt{24985}}{40}+\frac{15}{8}$$