Solve 2x+1/1+x^2=9/17

$$\frac { 2 x + 1 } { 1 + x ^ { 2 } } = \frac { 9 } { 17 }$$

Solve for x

$$x=4$$

$$x=-\frac{2}{9}\approx -0.222222222$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 17\left(x^{2}+1\right), the least common multiple of 1+x^{2},17.

$$17\left(2x+1\right)=9\left(x^{2}+1\right)$$

Use the distributive property to multiply 17 by 2x+1.

$$34x+17=9\left(x^{2}+1\right)$$

Use the distributive property to multiply 9 by x^{2}+1.

$$34x+17=9x^{2}+9$$

Subtract 9x^{2} from both sides.

$$34x+17-9x^{2}=9$$

Subtract 9 from both sides.

$$34x+17-9x^{2}-9=0$$

Subtract 9 from 17 to get 8.

$$34x+8-9x^{2}=0$$

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

$$-9x^{2}+34x+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=34 ab=-9\times 8=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 34.

$$a=36 b=-2$$

Rewrite -9x^{2}+34x+8 as \left(-9x^{2}+36x\right)+\left(-2x+8\right).

$$\left(-9x^{2}+36x\right)+\left(-2x+8\right)$$

Factor out 9x in the first and 2 in the second group.

$$9x\left(-x+4\right)+2\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(9x+2\right)$$

To find equation solutions, solve -x+4=0 and 9x+2=0.

$$x=4 x=-\frac{2}{9}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 17\left(x^{2}+1\right), the least common multiple of 1+x^{2},17.

$$17\left(2x+1\right)=9\left(x^{2}+1\right)$$

Use the distributive property to multiply 17 by 2x+1.

$$34x+17=9\left(x^{2}+1\right)$$

Use the distributive property to multiply 9 by x^{2}+1.

$$34x+17=9x^{2}+9$$

Subtract 9x^{2} from both sides.

$$34x+17-9x^{2}=9$$

Subtract 9 from both sides.

$$34x+17-9x^{2}-9=0$$

Subtract 9 from 17 to get 8.

$$34x+8-9x^{2}=0$$

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.

$$-9x^{2}+34x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 34 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-9\right)\times 8}}{2\left(-9\right)}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-9\right)\times 8}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-34±\sqrt{1156+36\times 8}}{2\left(-9\right)}$$

Multiply 36 times 8.

$$x=\frac{-34±\sqrt{1156+288}}{2\left(-9\right)}$$

Add 1156 to 288.

$$x=\frac{-34±\sqrt{1444}}{2\left(-9\right)}$$

Take the square root of 1444.

$$x=\frac{-34±38}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-34±38}{-18}$$

Now solve the equation x=\frac{-34±38}{-18} when ± is plus. Add -34 to 38.

$$x=\frac{4}{-18}$$

Reduce the fraction \frac{4}{-18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{9}$$

Now solve the equation x=\frac{-34±38}{-18} when ± is minus. Subtract 38 from -34.

$$x=-\frac{72}{-18}$$

Divide -72 by -18.

$$x=4$$

The equation is now solved.

$$x=-\frac{2}{9} x=4$$

Steps for Completing the Square

Multiply both sides of the equation by 17\left(x^{2}+1\right), the least common multiple of 1+x^{2},17.

$$17\left(2x+1\right)=9\left(x^{2}+1\right)$$

Use the distributive property to multiply 17 by 2x+1.

$$34x+17=9\left(x^{2}+1\right)$$

Use the distributive property to multiply 9 by x^{2}+1.

$$34x+17=9x^{2}+9$$

Subtract 9x^{2} from both sides.

$$34x+17-9x^{2}=9$$

Subtract 17 from both sides.

$$34x-9x^{2}=9-17$$

Subtract 17 from 9 to get -8.

$$34x-9x^{2}=-8$$

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.

$$-9x^{2}+34x=-8$$

Divide both sides by -9.

$$\frac{-9x^{2}+34x}{-9}=-\frac{8}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{34}{-9}x=-\frac{8}{-9}$$

Divide 34 by -9.

$$x^{2}-\frac{34}{9}x=-\frac{8}{-9}$$

Divide -8 by -9.

$$x^{2}-\frac{34}{9}x=\frac{8}{9}$$

Divide -\frac{34}{9}, the coefficient of the x term, by 2 to get -\frac{17}{9}. Then add the square of -\frac{17}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{34}{9}x+\left(-\frac{17}{9}\right)^{2}=\frac{8}{9}+\left(-\frac{17}{9}\right)^{2}$$

Square -\frac{17}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{34}{9}x+\frac{289}{81}=\frac{8}{9}+\frac{289}{81}$$

Add \frac{8}{9} to \frac{289}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{34}{9}x+\frac{289}{81}=\frac{361}{81}$$

Factor x^{2}-\frac{34}{9}x+\frac{289}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{17}{9}\right)^{2}=\frac{361}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{9}\right)^{2}}=\sqrt{\frac{361}{81}}$$

Simplify.

$$x-\frac{17}{9}=\frac{19}{9} x-\frac{17}{9}=-\frac{19}{9}$$

Add \frac{17}{9} to both sides of the equation.

$$x=4 x=-\frac{2}{9}$$