Solve 2/x=4x+7/5x^2+1

$$\frac{ 2 }{ x } = \frac{ 4x+7 }{ 5 { x }^{ 2 } +1 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=\frac{2}{3}\approx 0.666666667$$

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\left(5x^{2}+1\right), the least common multiple of x,5x^{2}+1.

$$\left(5x^{2}+1\right)\times 2=x\left(4x+7\right)$$

Use the distributive property to multiply 5x^{2}+1 by 2.

$$10x^{2}+2=x\left(4x+7\right)$$

Use the distributive property to multiply x by 4x+7.

$$10x^{2}+2=4x^{2}+7x$$

Subtract 4x^{2} from both sides.

$$10x^{2}+2-4x^{2}=7x$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

$$6x^{2}+2=7x$$

Subtract 7x from both sides.

$$6x^{2}+2-7x=0$$

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

$$6x^{2}-7x+2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-7 ab=6\times 2=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-4 b=-3$$

Rewrite 6x^{2}-7x+2 as \left(6x^{2}-4x\right)+\left(-3x+2\right).

$$\left(6x^{2}-4x\right)+\left(-3x+2\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(3x-2\right)-\left(3x-2\right)$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(2x-1\right)$$

To find equation solutions, solve 3x-2=0 and 2x-1=0.

$$x=\frac{2}{3} x=\frac{1}{2}$$

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\left(5x^{2}+1\right), the least common multiple of x,5x^{2}+1.

$$\left(5x^{2}+1\right)\times 2=x\left(4x+7\right)$$

Use the distributive property to multiply 5x^{2}+1 by 2.

$$10x^{2}+2=x\left(4x+7\right)$$

Use the distributive property to multiply x by 4x+7.

$$10x^{2}+2=4x^{2}+7x$$

Subtract 4x^{2} from both sides.

$$10x^{2}+2-4x^{2}=7x$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

$$6x^{2}+2=7x$$

Subtract 7x from both sides.

$$6x^{2}+2-7x=0$$

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.

$$6x^{2}-7x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -7 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\times 2}}{2\times 6}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\times 2}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\times 2}}{2\times 6}$$

Multiply -24 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-48}}{2\times 6}$$

Add 49 to -48.

$$x=\frac{-\left(-7\right)±\sqrt{1}}{2\times 6}$$

Take the square root of 1.

$$x=\frac{-\left(-7\right)±1}{2\times 6}$$

The opposite of -7 is 7.

$$x=\frac{7±1}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{7±1}{12}$$

Now solve the equation x=\frac{7±1}{12} when ± is plus. Add 7 to 1.

$$x=\frac{8}{12}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{7±1}{12} when ± is minus. Subtract 1 from 7.

$$x=\frac{6}{12}$$

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=\frac{2}{3} x=\frac{1}{2}$$

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\left(5x^{2}+1\right), the least common multiple of x,5x^{2}+1.

$$\left(5x^{2}+1\right)\times 2=x\left(4x+7\right)$$

Use the distributive property to multiply 5x^{2}+1 by 2.

$$10x^{2}+2=x\left(4x+7\right)$$

Use the distributive property to multiply x by 4x+7.

$$10x^{2}+2=4x^{2}+7x$$

Subtract 4x^{2} from both sides.

$$10x^{2}+2-4x^{2}=7x$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

$$6x^{2}+2=7x$$

Subtract 7x from both sides.

$$6x^{2}+2-7x=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}-7x=-2$$

Divide both sides by 6.

$$\frac{6x^{2}-7x}{6}=-\frac{2}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{7}{6}x=-\frac{2}{6}$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{6}x=-\frac{1}{3}$$

Divide -\frac{7}{6}, the coefficient of the x term, by 2 to get -\frac{7}{12}. Then add the square of -\frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=-\frac{1}{3}+\left(-\frac{7}{12}\right)^{2}$$

Square -\frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{1}{3}+\frac{49}{144}$$

Add -\frac{1}{3} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=\frac{1}{144}$$

Factor x^{2}-\frac{7}{6}x+\frac{49}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$x-\frac{7}{12}=\frac{1}{12} x-\frac{7}{12}=-\frac{1}{12}$$

Add \frac{7}{12} to both sides of the equation.

$$x=\frac{2}{3} x=\frac{1}{2}$$