Solve 15/7+x+15/4-x=842/3

$$\frac { 15 } { 7 + x } + \frac { 15 } { 4 - x } = 84 \frac { 2 } { 3 }$$

Solve for x

$$x=\frac{\sqrt{1825879}}{254}-\frac{3}{2}\approx 3.819885441$$

$$x=-\frac{\sqrt{1825879}}{254}-\frac{3}{2}\approx -6.819885441$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -7,4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-4\right)\left(x+7\right), the least common multiple of 7+x,4-x,3.

$$\left(3x-12\right)\times 15+\left(-21-3x\right)\times 15=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Use the distributive property to multiply 3x-12 by 15.

$$45x-180+\left(-21-3x\right)\times 15=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Use the distributive property to multiply -21-3x by 15.

$$45x-180-315-45x=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Subtract 315 from -180 to get -495.

$$45x-495-45x=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Combine 45x and -45x to get 0.

$$-495=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Multiply 84 and 3 to get 252.

$$-495=\left(x-4\right)\left(x+7\right)\left(252+2\right)$$

Add 252 and 2 to get 254.

$$-495=\left(x-4\right)\left(x+7\right)\times 254$$

Use the distributive property to multiply x-4 by x+7 and combine like terms.

$$-495=\left(x^{2}+3x-28\right)\times 254$$

Use the distributive property to multiply x^{2}+3x-28 by 254.

$$-495=254x^{2}+762x-7112$$

Swap sides so that all variable terms are on the left hand side.

$$254x^{2}+762x-7112=-495$$

Add 495 to both sides.

$$254x^{2}+762x-7112+495=0$$

Add -7112 and 495 to get -6617.

$$254x^{2}+762x-6617=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 254 for a, 762 for b, and -6617 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-762±\sqrt{762^{2}-4\times 254\left(-6617\right)}}{2\times 254}$$

Square 762.

$$x=\frac{-762±\sqrt{580644-4\times 254\left(-6617\right)}}{2\times 254}$$

Multiply -4 times 254.

$$x=\frac{-762±\sqrt{580644-1016\left(-6617\right)}}{2\times 254}$$

Multiply -1016 times -6617.

$$x=\frac{-762±\sqrt{580644+6722872}}{2\times 254}$$

Add 580644 to 6722872.

$$x=\frac{-762±\sqrt{7303516}}{2\times 254}$$

Take the square root of 7303516.

$$x=\frac{-762±2\sqrt{1825879}}{2\times 254}$$

Multiply 2 times 254.

$$x=\frac{-762±2\sqrt{1825879}}{508}$$

Now solve the equation x=\frac{-762±2\sqrt{1825879}}{508} when ± is plus. Add -762 to 2\sqrt{1825879}.

$$x=\frac{2\sqrt{1825879}-762}{508}$$

Divide -762+2\sqrt{1825879} by 508.

$$x=\frac{\sqrt{1825879}}{254}-\frac{3}{2}$$

Now solve the equation x=\frac{-762±2\sqrt{1825879}}{508} when ± is minus. Subtract 2\sqrt{1825879} from -762.

$$x=\frac{-2\sqrt{1825879}-762}{508}$$

Divide -762-2\sqrt{1825879} by 508.

$$x=-\frac{\sqrt{1825879}}{254}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1825879}}{254}-\frac{3}{2} x=-\frac{\sqrt{1825879}}{254}-\frac{3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -7,4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-4\right)\left(x+7\right), the least common multiple of 7+x,4-x,3.

$$\left(3x-12\right)\times 15+\left(-21-3x\right)\times 15=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Use the distributive property to multiply 3x-12 by 15.

$$45x-180+\left(-21-3x\right)\times 15=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Use the distributive property to multiply -21-3x by 15.

$$45x-180-315-45x=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Subtract 315 from -180 to get -495.

$$45x-495-45x=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Combine 45x and -45x to get 0.

$$-495=\left(x-4\right)\left(x+7\right)\left(84\times 3+2\right)$$

Multiply 84 and 3 to get 252.

$$-495=\left(x-4\right)\left(x+7\right)\left(252+2\right)$$

Add 252 and 2 to get 254.

$$-495=\left(x-4\right)\left(x+7\right)\times 254$$

Use the distributive property to multiply x-4 by x+7 and combine like terms.

$$-495=\left(x^{2}+3x-28\right)\times 254$$

Use the distributive property to multiply x^{2}+3x-28 by 254.

$$-495=254x^{2}+762x-7112$$

Swap sides so that all variable terms are on the left hand side.

$$254x^{2}+762x-7112=-495$$

Add 7112 to both sides.

$$254x^{2}+762x=-495+7112$$

Add -495 and 7112 to get 6617.

$$254x^{2}+762x=6617$$

Divide both sides by 254.

$$\frac{254x^{2}+762x}{254}=\frac{6617}{254}$$

Dividing by 254 undoes the multiplication by 254.

$$x^{2}+\frac{762}{254}x=\frac{6617}{254}$$

Divide 762 by 254.

$$x^{2}+3x=\frac{6617}{254}$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{6617}{254}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=\frac{6617}{254}+\frac{9}{4}$$

Add \frac{6617}{254} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{14377}{508}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{14377}{508}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{14377}{508}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{1825879}}{254} x+\frac{3}{2}=-\frac{\sqrt{1825879}}{254}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{1825879}}{254}-\frac{3}{2} x=-\frac{\sqrt{1825879}}{254}-\frac{3}{2}$$