Solve 27/x^2+x+3/2x=4

$$\frac { 27 } { x ^ { 2 } } + \frac { x + 3 } { 2 x } = 4$$

Solve for x

$$x = -\frac{18}{7} = -2\frac{4}{7} \approx -2.571428571$$

$$x=3$$

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 2x^{2}, the least common multiple of x^{2},2x.

$$2\times 27+x\left(x+3\right)=8x^{2}$$

Multiply 2 and 27 to get 54.

$$54+x\left(x+3\right)=8x^{2}$$

Use the distributive property to multiply x by x+3.

$$54+x^{2}+3x=8x^{2}$$

Subtract 8x^{2} from both sides.

$$54+x^{2}+3x-8x^{2}=0$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$54-7x^{2}+3x=0$$

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

$$-7x^{2}+3x+54=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx+54. To find a and b, set up a system to be solved.

$$a+b=3 ab=-7\times 54=-378$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -378.

$$-1,378 -2,189 -3,126 -6,63 -7,54 -9,42 -14,27 -18,21$$

Calculate the sum for each pair.

$$-1+378=377 -2+189=187 -3+126=123 -6+63=57 -7+54=47 -9+42=33 -14+27=13 -18+21=3$$

The solution is the pair that gives sum 3.

$$a=21 b=-18$$

Rewrite -7x^{2}+3x+54 as \left(-7x^{2}+21x\right)+\left(-18x+54\right).

$$\left(-7x^{2}+21x\right)+\left(-18x+54\right)$$

Factor out 7x in the first and 18 in the second group.

$$7x\left(-x+3\right)+18\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(7x+18\right)$$

To find equation solutions, solve -x+3=0 and 7x+18=0.

$$x=3 x=-\frac{18}{7}$$

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 2x^{2}, the least common multiple of x^{2},2x.

$$2\times 27+x\left(x+3\right)=8x^{2}$$

Multiply 2 and 27 to get 54.

$$54+x\left(x+3\right)=8x^{2}$$

Use the distributive property to multiply x by x+3.

$$54+x^{2}+3x=8x^{2}$$

Subtract 8x^{2} from both sides.

$$54+x^{2}+3x-8x^{2}=0$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$54-7x^{2}+3x=0$$

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.

$$-7x^{2}+3x+54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 3 for b, and 54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-7\right)\times 54}}{2\left(-7\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-7\right)\times 54}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-3±\sqrt{9+28\times 54}}{2\left(-7\right)}$$

Multiply 28 times 54.

$$x=\frac{-3±\sqrt{9+1512}}{2\left(-7\right)}$$

Add 9 to 1512.

$$x=\frac{-3±\sqrt{1521}}{2\left(-7\right)}$$

Take the square root of 1521.

$$x=\frac{-3±39}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-3±39}{-14}$$

Now solve the equation x=\frac{-3±39}{-14} when ± is plus. Add -3 to 39.

$$x=\frac{36}{-14}$$

Reduce the fraction \frac{36}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{18}{7}$$

Now solve the equation x=\frac{-3±39}{-14} when ± is minus. Subtract 39 from -3.

$$x=-\frac{42}{-14}$$

Divide -42 by -14.

$$x=3$$

The equation is now solved.

$$x=-\frac{18}{7} x=3$$

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 2x^{2}, the least common multiple of x^{2},2x.

$$2\times 27+x\left(x+3\right)=8x^{2}$$

Multiply 2 and 27 to get 54.

$$54+x\left(x+3\right)=8x^{2}$$

Use the distributive property to multiply x by x+3.

$$54+x^{2}+3x=8x^{2}$$

Subtract 8x^{2} from both sides.

$$54+x^{2}+3x-8x^{2}=0$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$54-7x^{2}+3x=0$$

Subtract 54 from both sides. Anything subtracted from zero gives its negation.

$$-7x^{2}+3x=-54$$

Divide both sides by -7.

$$\frac{-7x^{2}+3x}{-7}=-\frac{54}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{3}{-7}x=-\frac{54}{-7}$$

Divide 3 by -7.

$$x^{2}-\frac{3}{7}x=-\frac{54}{-7}$$

Divide -54 by -7.

$$x^{2}-\frac{3}{7}x=\frac{54}{7}$$

Divide -\frac{3}{7}, the coefficient of the x term, by 2 to get -\frac{3}{14}. Then add the square of -\frac{3}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=\frac{54}{7}+\left(-\frac{3}{14}\right)^{2}$$

Square -\frac{3}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{54}{7}+\frac{9}{196}$$

Add \frac{54}{7} to \frac{9}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{1521}{196}$$

Factor x^{2}-\frac{3}{7}x+\frac{9}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{14}\right)^{2}=\frac{1521}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{\frac{1521}{196}}$$

Simplify.

$$x-\frac{3}{14}=\frac{39}{14} x-\frac{3}{14}=-\frac{39}{14}$$

Add \frac{3}{14} to both sides of the equation.

$$x=3 x=-\frac{18}{7}$$