Solve 45/x-2+15/x+4=14

$$\frac { 45 } { x - 2 } + \frac { 15 } { x + 4 } = 14$$

Solve for x

$$x = \frac{3 \sqrt{109} + 8}{7} \approx 5.617274218$$

$$x=\frac{8-3\sqrt{109}}{7}\approx -3.331559932$$

Steps Using the Quadratic Formula

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

$$\left(x+4\right)\times 45+\left(x-2\right)\times 15=14\left(x-2\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by 45.

$$45x+180+\left(x-2\right)\times 15=14\left(x-2\right)\left(x+4\right)$$

Use the distributive property to multiply x-2 by 15.

$$45x+180+15x-30=14\left(x-2\right)\left(x+4\right)$$

Combine 45x and 15x to get 60x.

$$60x+180-30=14\left(x-2\right)\left(x+4\right)$$

Subtract 30 from 180 to get 150.

$$60x+150=14\left(x-2\right)\left(x+4\right)$$

Use the distributive property to multiply 14 by x-2.

$$60x+150=\left(14x-28\right)\left(x+4\right)$$

Use the distributive property to multiply 14x-28 by x+4 and combine like terms.

$$60x+150=14x^{2}+28x-112$$

Subtract 14x^{2} from both sides.

$$60x+150-14x^{2}=28x-112$$

Subtract 28x from both sides.

$$60x+150-14x^{2}-28x=-112$$

Combine 60x and -28x to get 32x.

$$32x+150-14x^{2}=-112$$

Add 112 to both sides.

$$32x+150-14x^{2}+112=0$$

Add 150 and 112 to get 262.

$$32x+262-14x^{2}=0$$

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.

$$-14x^{2}+32x+262=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, 32 for b, and 262 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-14\right)\times 262}}{2\left(-14\right)}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-14\right)\times 262}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-32±\sqrt{1024+56\times 262}}{2\left(-14\right)}$$

Multiply 56 times 262.

$$x=\frac{-32±\sqrt{1024+14672}}{2\left(-14\right)}$$

Add 1024 to 14672.

$$x=\frac{-32±\sqrt{15696}}{2\left(-14\right)}$$

Take the square root of 15696.

$$x=\frac{-32±12\sqrt{109}}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-32±12\sqrt{109}}{-28}$$

Now solve the equation x=\frac{-32±12\sqrt{109}}{-28} when ± is plus. Add -32 to 12\sqrt{109}.

$$x=\frac{12\sqrt{109}-32}{-28}$$

Divide -32+12\sqrt{109} by -28.

$$x=\frac{8-3\sqrt{109}}{7}$$

Now solve the equation x=\frac{-32±12\sqrt{109}}{-28} when ± is minus. Subtract 12\sqrt{109} from -32.

$$x=\frac{-12\sqrt{109}-32}{-28}$$

Divide -32-12\sqrt{109} by -28.

$$x=\frac{3\sqrt{109}+8}{7}$$

The equation is now solved.

$$x=\frac{8-3\sqrt{109}}{7} x=\frac{3\sqrt{109}+8}{7}$$

Steps for Completing the Square

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

$$\left(x+4\right)\times 45+\left(x-2\right)\times 15=14\left(x-2\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by 45.

$$45x+180+\left(x-2\right)\times 15=14\left(x-2\right)\left(x+4\right)$$

Use the distributive property to multiply x-2 by 15.

$$45x+180+15x-30=14\left(x-2\right)\left(x+4\right)$$

Combine 45x and 15x to get 60x.

$$60x+180-30=14\left(x-2\right)\left(x+4\right)$$

Subtract 30 from 180 to get 150.

$$60x+150=14\left(x-2\right)\left(x+4\right)$$

Use the distributive property to multiply 14 by x-2.

$$60x+150=\left(14x-28\right)\left(x+4\right)$$

Use the distributive property to multiply 14x-28 by x+4 and combine like terms.

$$60x+150=14x^{2}+28x-112$$

Subtract 14x^{2} from both sides.

$$60x+150-14x^{2}=28x-112$$

Subtract 28x from both sides.

$$60x+150-14x^{2}-28x=-112$$

Combine 60x and -28x to get 32x.

$$32x+150-14x^{2}=-112$$

Subtract 150 from both sides.

$$32x-14x^{2}=-112-150$$

Subtract 150 from -112 to get -262.

$$32x-14x^{2}=-262$$

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.

$$-14x^{2}+32x=-262$$

Divide both sides by -14.

$$\frac{-14x^{2}+32x}{-14}=-\frac{262}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$x^{2}+\frac{32}{-14}x=-\frac{262}{-14}$$

Reduce the fraction \frac{32}{-14} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{16}{7}x=-\frac{262}{-14}$$

Reduce the fraction \frac{-262}{-14} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{16}{7}x=\frac{131}{7}$$

Divide -\frac{16}{7}, the coefficient of the x term, by 2 to get -\frac{8}{7}. Then add the square of -\frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{7}x+\left(-\frac{8}{7}\right)^{2}=\frac{131}{7}+\left(-\frac{8}{7}\right)^{2}$$

Square -\frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{7}x+\frac{64}{49}=\frac{131}{7}+\frac{64}{49}$$

Add \frac{131}{7} to \frac{64}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{7}x+\frac{64}{49}=\frac{981}{49}$$

Factor x^{2}-\frac{16}{7}x+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{7}\right)^{2}=\frac{981}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{7}\right)^{2}}=\sqrt{\frac{981}{49}}$$

Simplify.

$$x-\frac{8}{7}=\frac{3\sqrt{109}}{7} x-\frac{8}{7}=-\frac{3\sqrt{109}}{7}$$

Add \frac{8}{7} to both sides of the equation.

$$x=\frac{3\sqrt{109}+8}{7} x=\frac{8-3\sqrt{109}}{7}$$