Solve 4/x^2-55/x=14

$$\frac { 4 } { x ^ { 2 } } - \frac { 55 } { x } = 14$$

Solve for x

$$x=-4$$

$$x=\frac{1}{14}\approx 0.071428571$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.

$$4-x\times 55=14x^{2}$$

Subtract 14x^{2} from both sides.

$$4-x\times 55-14x^{2}=0$$

Multiply -1 and 55 to get -55.

$$4-55x-14x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-14x^{2}-55x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -14x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-55 ab=-14\times 4=-56$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -56.

$$1,-56 2,-28 4,-14 7,-8$$

Calculate the sum for each pair.

$$1-56=-55 2-28=-26 4-14=-10 7-8=-1$$

The solution is the pair that gives sum -55.

$$a=1 b=-56$$

Rewrite -14x^{2}-55x+4 as \left(-14x^{2}+x\right)+\left(-56x+4\right).

$$\left(-14x^{2}+x\right)+\left(-56x+4\right)$$

Factor out -x in the first and -4 in the second group.

$$-x\left(14x-1\right)-4\left(14x-1\right)$$

Factor out common term 14x-1 by using distributive property.

$$\left(14x-1\right)\left(-x-4\right)$$

To find equation solutions, solve 14x-1=0 and -x-4=0.

$$x=\frac{1}{14} x=-4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.

$$4-x\times 55=14x^{2}$$

Subtract 14x^{2} from both sides.

$$4-x\times 55-14x^{2}=0$$

Multiply -1 and 55 to get -55.

$$4-55x-14x^{2}=0$$

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.

$$-14x^{2}-55x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, -55 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Square -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Multiply -4 times -14.

$$x=\frac{-\left(-55\right)±\sqrt{3025+56\times 4}}{2\left(-14\right)}$$

Multiply 56 times 4.

$$x=\frac{-\left(-55\right)±\sqrt{3025+224}}{2\left(-14\right)}$$

Add 3025 to 224.

$$x=\frac{-\left(-55\right)±\sqrt{3249}}{2\left(-14\right)}$$

Take the square root of 3249.

$$x=\frac{-\left(-55\right)±57}{2\left(-14\right)}$$

The opposite of -55 is 55.

$$x=\frac{55±57}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{55±57}{-28}$$

Now solve the equation x=\frac{55±57}{-28} when ± is plus. Add 55 to 57.

$$x=\frac{112}{-28}$$

Divide 112 by -28.

$$x=-4$$

Now solve the equation x=\frac{55±57}{-28} when ± is minus. Subtract 57 from 55.

$$x=-\frac{2}{-28}$$

Reduce the fraction \frac{-2}{-28} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{14}$$

The equation is now solved.

$$x=-4 x=\frac{1}{14}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x^{2},x.

$$4-x\times 55=14x^{2}$$

Subtract 14x^{2} from both sides.

$$4-x\times 55-14x^{2}=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-x\times 55-14x^{2}=-4$$

Multiply -1 and 55 to get -55.

$$-55x-14x^{2}=-4$$

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.

$$-14x^{2}-55x=-4$$

Divide both sides by -14.

$$\frac{-14x^{2}-55x}{-14}=-\frac{4}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$x^{2}+\left(-\frac{55}{-14}\right)x=-\frac{4}{-14}$$

Divide -55 by -14.

$$x^{2}+\frac{55}{14}x=-\frac{4}{-14}$$

Reduce the fraction \frac{-4}{-14} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{55}{14}x=\frac{2}{7}$$

Divide \frac{55}{14}, the coefficient of the x term, by 2 to get \frac{55}{28}. Then add the square of \frac{55}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{55}{14}x+\left(\frac{55}{28}\right)^{2}=\frac{2}{7}+\left(\frac{55}{28}\right)^{2}$$

Square \frac{55}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{55}{14}x+\frac{3025}{784}=\frac{2}{7}+\frac{3025}{784}$$

Add \frac{2}{7} to \frac{3025}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{55}{14}x+\frac{3025}{784}=\frac{3249}{784}$$

Factor x^{2}+\frac{55}{14}x+\frac{3025}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{55}{28}\right)^{2}=\frac{3249}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{55}{28}\right)^{2}}=\sqrt{\frac{3249}{784}}$$

Simplify.

$$x+\frac{55}{28}=\frac{57}{28} x+\frac{55}{28}=-\frac{57}{28}$$

Subtract \frac{55}{28} from both sides of the equation.

$$x=\frac{1}{14} x=-4$$