Solve 2x/x^2+1=7/25

$$\frac { 2 x } { x ^ { 2 } + 1 } = \frac { 7 } { 25 }$$

Solve for x

$$x=7$$

$$x=\frac{1}{7}\approx 0.142857143$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 25\left(x^{2}+1\right), the least common multiple of x^{2}+1,25.

$$25\times 2x=7\left(x^{2}+1\right)$$

Multiply 25 and 2 to get 50.

$$50x=7\left(x^{2}+1\right)$$

Use the distributive property to multiply 7 by x^{2}+1.

$$50x=7x^{2}+7$$

Subtract 7x^{2} from both sides.

$$50x-7x^{2}=7$$

Subtract 7 from both sides.

$$50x-7x^{2}-7=0$$

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

$$-7x^{2}+50x-7=0$$

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

$$a+b=50 ab=-7\left(-7\right)=49$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 49.

$$1,49 7,7$$

Calculate the sum for each pair.

$$1+49=50 7+7=14$$

The solution is the pair that gives sum 50.

$$a=49 b=1$$

Rewrite -7x^{2}+50x-7 as \left(-7x^{2}+49x\right)+\left(x-7\right).

$$\left(-7x^{2}+49x\right)+\left(x-7\right)$$

Factor out 7x in the first and -1 in the second group.

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

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(7x-1\right)$$

To find equation solutions, solve -x+7=0 and 7x-1=0.

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

Steps Using the Quadratic Formula

Multiply both sides of the equation by 25\left(x^{2}+1\right), the least common multiple of x^{2}+1,25.

$$25\times 2x=7\left(x^{2}+1\right)$$

Multiply 25 and 2 to get 50.

$$50x=7\left(x^{2}+1\right)$$

Use the distributive property to multiply 7 by x^{2}+1.

$$50x=7x^{2}+7$$

Subtract 7x^{2} from both sides.

$$50x-7x^{2}=7$$

Subtract 7 from both sides.

$$50x-7x^{2}-7=0$$

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.

$$-7x^{2}+50x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 50 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-7\right)\left(-7\right)}}{2\left(-7\right)}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-7\right)\left(-7\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-50±\sqrt{2500+28\left(-7\right)}}{2\left(-7\right)}$$

Multiply 28 times -7.

$$x=\frac{-50±\sqrt{2500-196}}{2\left(-7\right)}$$

Add 2500 to -196.

$$x=\frac{-50±\sqrt{2304}}{2\left(-7\right)}$$

Take the square root of 2304.

$$x=\frac{-50±48}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-50±48}{-14}$$

Now solve the equation x=\frac{-50±48}{-14} when ± is plus. Add -50 to 48.

$$x=-\frac{2}{-14}$$

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

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

Now solve the equation x=\frac{-50±48}{-14} when ± is minus. Subtract 48 from -50.

$$x=-\frac{98}{-14}$$

Divide -98 by -14.

$$x=7$$

The equation is now solved.

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

Steps for Completing the Square

Multiply both sides of the equation by 25\left(x^{2}+1\right), the least common multiple of x^{2}+1,25.

$$25\times 2x=7\left(x^{2}+1\right)$$

Multiply 25 and 2 to get 50.

$$50x=7\left(x^{2}+1\right)$$

Use the distributive property to multiply 7 by x^{2}+1.

$$50x=7x^{2}+7$$

Subtract 7x^{2} from both sides.

$$50x-7x^{2}=7$$

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.

$$-7x^{2}+50x=7$$

Divide both sides by -7.

$$\frac{-7x^{2}+50x}{-7}=\frac{7}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{50}{-7}x=\frac{7}{-7}$$

Divide 50 by -7.

$$x^{2}-\frac{50}{7}x=\frac{7}{-7}$$

Divide 7 by -7.

$$x^{2}-\frac{50}{7}x=-1$$

Divide -\frac{50}{7}, the coefficient of the x term, by 2 to get -\frac{25}{7}. Then add the square of -\frac{25}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{50}{7}x+\left(-\frac{25}{7}\right)^{2}=-1+\left(-\frac{25}{7}\right)^{2}$$

Square -\frac{25}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{50}{7}x+\frac{625}{49}=-1+\frac{625}{49}$$

Add -1 to \frac{625}{49}.

$$x^{2}-\frac{50}{7}x+\frac{625}{49}=\frac{576}{49}$$

Factor x^{2}-\frac{50}{7}x+\frac{625}{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}.

$$\left(x-\frac{25}{7}\right)^{2}=\frac{576}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{7}\right)^{2}}=\sqrt{\frac{576}{49}}$$

Simplify.

$$x-\frac{25}{7}=\frac{24}{7} x-\frac{25}{7}=-\frac{24}{7}$$

Add \frac{25}{7} to both sides of the equation.

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