Solve 3/x+2-9/7x-14=15

$$\frac { 3 } { x + 2 } - \frac { 9 } { 7 x - 14 } = 15$$

Solve for x

$$x = \frac{2 \sqrt{1051} + 2}{35} \approx 1.90966458$$

$$x=\frac{2-2\sqrt{1051}}{35}\approx -1.795378866$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 7\left(x-2\right)\left(x+2\right), the least common multiple of x+2,7x-14.

$$\left(7x-14\right)\times 3-\left(x+2\right)\times 9=105\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 7x-14 by 3.

$$21x-42-\left(x+2\right)\times 9=105\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 9.

$$21x-42-\left(9x+18\right)=105\left(x-2\right)\left(x+2\right)$$

To find the opposite of 9x+18, find the opposite of each term.

$$21x-42-9x-18=105\left(x-2\right)\left(x+2\right)$$

Combine 21x and -9x to get 12x.

$$12x-42-18=105\left(x-2\right)\left(x+2\right)$$

Subtract 18 from -42 to get -60.

$$12x-60=105\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 105 by x-2.

$$12x-60=\left(105x-210\right)\left(x+2\right)$$

Use the distributive property to multiply 105x-210 by x+2 and combine like terms.

$$12x-60=105x^{2}-420$$

Subtract 105x^{2} from both sides.

$$12x-60-105x^{2}=-420$$

Add 420 to both sides.

$$12x-60-105x^{2}+420=0$$

Add -60 and 420 to get 360.

$$12x+360-105x^{2}=0$$

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.

$$-105x^{2}+12x+360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -105 for a, 12 for b, and 360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-105\right)\times 360}}{2\left(-105\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-105\right)\times 360}}{2\left(-105\right)}$$

Multiply -4 times -105.

$$x=\frac{-12±\sqrt{144+420\times 360}}{2\left(-105\right)}$$

Multiply 420 times 360.

$$x=\frac{-12±\sqrt{144+151200}}{2\left(-105\right)}$$

Add 144 to 151200.

$$x=\frac{-12±\sqrt{151344}}{2\left(-105\right)}$$

Take the square root of 151344.

$$x=\frac{-12±12\sqrt{1051}}{2\left(-105\right)}$$

Multiply 2 times -105.

$$x=\frac{-12±12\sqrt{1051}}{-210}$$

Now solve the equation x=\frac{-12±12\sqrt{1051}}{-210} when ± is plus. Add -12 to 12\sqrt{1051}.

$$x=\frac{12\sqrt{1051}-12}{-210}$$

Divide -12+12\sqrt{1051} by -210.

$$x=\frac{2-2\sqrt{1051}}{35}$$

Now solve the equation x=\frac{-12±12\sqrt{1051}}{-210} when ± is minus. Subtract 12\sqrt{1051} from -12.

$$x=\frac{-12\sqrt{1051}-12}{-210}$$

Divide -12-12\sqrt{1051} by -210.

$$x=\frac{2\sqrt{1051}+2}{35}$$

The equation is now solved.

$$x=\frac{2-2\sqrt{1051}}{35} x=\frac{2\sqrt{1051}+2}{35}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 7\left(x-2\right)\left(x+2\right), the least common multiple of x+2,7x-14.

$$\left(7x-14\right)\times 3-\left(x+2\right)\times 9=105\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 7x-14 by 3.

$$21x-42-\left(x+2\right)\times 9=105\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 9.

$$21x-42-\left(9x+18\right)=105\left(x-2\right)\left(x+2\right)$$

To find the opposite of 9x+18, find the opposite of each term.

$$21x-42-9x-18=105\left(x-2\right)\left(x+2\right)$$

Combine 21x and -9x to get 12x.

$$12x-42-18=105\left(x-2\right)\left(x+2\right)$$

Subtract 18 from -42 to get -60.

$$12x-60=105\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 105 by x-2.

$$12x-60=\left(105x-210\right)\left(x+2\right)$$

Use the distributive property to multiply 105x-210 by x+2 and combine like terms.

$$12x-60=105x^{2}-420$$

Subtract 105x^{2} from both sides.

$$12x-60-105x^{2}=-420$$

Add 60 to both sides.

$$12x-105x^{2}=-420+60$$

Add -420 and 60 to get -360.

$$12x-105x^{2}=-360$$

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.

$$-105x^{2}+12x=-360$$

Divide both sides by -105.

$$\frac{-105x^{2}+12x}{-105}=-\frac{360}{-105}$$

Dividing by -105 undoes the multiplication by -105.

$$x^{2}+\frac{12}{-105}x=-\frac{360}{-105}$$

Reduce the fraction \frac{12}{-105} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{4}{35}x=-\frac{360}{-105}$$

Reduce the fraction \frac{-360}{-105} to lowest terms by extracting and canceling out 15.

$$x^{2}-\frac{4}{35}x=\frac{24}{7}$$

Divide -\frac{4}{35}, the coefficient of the x term, by 2 to get -\frac{2}{35}. Then add the square of -\frac{2}{35} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{35}x+\left(-\frac{2}{35}\right)^{2}=\frac{24}{7}+\left(-\frac{2}{35}\right)^{2}$$

Square -\frac{2}{35} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{35}x+\frac{4}{1225}=\frac{24}{7}+\frac{4}{1225}$$

Add \frac{24}{7} to \frac{4}{1225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{35}x+\frac{4}{1225}=\frac{4204}{1225}$$

Factor x^{2}-\frac{4}{35}x+\frac{4}{1225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{35}\right)^{2}=\frac{4204}{1225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{35}\right)^{2}}=\sqrt{\frac{4204}{1225}}$$

Simplify.

$$x-\frac{2}{35}=\frac{2\sqrt{1051}}{35} x-\frac{2}{35}=-\frac{2\sqrt{1051}}{35}$$

Add \frac{2}{35} to both sides of the equation.

$$x=\frac{2\sqrt{1051}+2}{35} x=\frac{2-2\sqrt{1051}}{35}$$