Solve 3/(2y-5)+2/y+7=17

$$\frac { 3 } { ( 2 y - 5 ) } + \frac { 2 } { y + 7 } = 17$$

Solve for y

$$y = \frac{\sqrt{25933} - 73}{34} \approx 2.589331258$$

$$y=\frac{-\sqrt{25933}-73}{34}\approx -6.883448905$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -7,\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2y-5\right)\left(y+7\right), the least common multiple of 2y-5,y+7.

$$\left(y+7\right)\times 3+\left(2y-5\right)\times 2=17\left(2y-5\right)\left(y+7\right)$$

Use the distributive property to multiply y+7 by 3.

$$3y+21+\left(2y-5\right)\times 2=17\left(2y-5\right)\left(y+7\right)$$

Use the distributive property to multiply 2y-5 by 2.

$$3y+21+4y-10=17\left(2y-5\right)\left(y+7\right)$$

Combine 3y and 4y to get 7y.

$$7y+21-10=17\left(2y-5\right)\left(y+7\right)$$

Subtract 10 from 21 to get 11.

$$7y+11=17\left(2y-5\right)\left(y+7\right)$$

Use the distributive property to multiply 17 by 2y-5.

$$7y+11=\left(34y-85\right)\left(y+7\right)$$

Use the distributive property to multiply 34y-85 by y+7 and combine like terms.

$$7y+11=34y^{2}+153y-595$$

Subtract 34y^{2} from both sides.

$$7y+11-34y^{2}=153y-595$$

Subtract 153y from both sides.

$$7y+11-34y^{2}-153y=-595$$

Combine 7y and -153y to get -146y.

$$-146y+11-34y^{2}=-595$$

Add 595 to both sides.

$$-146y+11-34y^{2}+595=0$$

Add 11 and 595 to get 606.

$$-146y+606-34y^{2}=0$$

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.

$$-34y^{2}-146y+606=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -34 for a, -146 for b, and 606 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-146\right)±\sqrt{\left(-146\right)^{2}-4\left(-34\right)\times 606}}{2\left(-34\right)}$$

Square -146.

$$y=\frac{-\left(-146\right)±\sqrt{21316-4\left(-34\right)\times 606}}{2\left(-34\right)}$$

Multiply -4 times -34.

$$y=\frac{-\left(-146\right)±\sqrt{21316+136\times 606}}{2\left(-34\right)}$$

Multiply 136 times 606.

$$y=\frac{-\left(-146\right)±\sqrt{21316+82416}}{2\left(-34\right)}$$

Add 21316 to 82416.

$$y=\frac{-\left(-146\right)±\sqrt{103732}}{2\left(-34\right)}$$

Take the square root of 103732.

$$y=\frac{-\left(-146\right)±2\sqrt{25933}}{2\left(-34\right)}$$

The opposite of -146 is 146.

$$y=\frac{146±2\sqrt{25933}}{2\left(-34\right)}$$

Multiply 2 times -34.

$$y=\frac{146±2\sqrt{25933}}{-68}$$

Now solve the equation y=\frac{146±2\sqrt{25933}}{-68} when ± is plus. Add 146 to 2\sqrt{25933}.

$$y=\frac{2\sqrt{25933}+146}{-68}$$

Divide 146+2\sqrt{25933} by -68.

$$y=\frac{-\sqrt{25933}-73}{34}$$

Now solve the equation y=\frac{146±2\sqrt{25933}}{-68} when ± is minus. Subtract 2\sqrt{25933} from 146.

$$y=\frac{146-2\sqrt{25933}}{-68}$$

Divide 146-2\sqrt{25933} by -68.

$$y=\frac{\sqrt{25933}-73}{34}$$

The equation is now solved.

$$y=\frac{-\sqrt{25933}-73}{34} y=\frac{\sqrt{25933}-73}{34}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -7,\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2y-5\right)\left(y+7\right), the least common multiple of 2y-5,y+7.

$$\left(y+7\right)\times 3+\left(2y-5\right)\times 2=17\left(2y-5\right)\left(y+7\right)$$

Use the distributive property to multiply y+7 by 3.

$$3y+21+\left(2y-5\right)\times 2=17\left(2y-5\right)\left(y+7\right)$$

Use the distributive property to multiply 2y-5 by 2.

$$3y+21+4y-10=17\left(2y-5\right)\left(y+7\right)$$

Combine 3y and 4y to get 7y.

$$7y+21-10=17\left(2y-5\right)\left(y+7\right)$$

Subtract 10 from 21 to get 11.

$$7y+11=17\left(2y-5\right)\left(y+7\right)$$

Use the distributive property to multiply 17 by 2y-5.

$$7y+11=\left(34y-85\right)\left(y+7\right)$$

Use the distributive property to multiply 34y-85 by y+7 and combine like terms.

$$7y+11=34y^{2}+153y-595$$

Subtract 34y^{2} from both sides.

$$7y+11-34y^{2}=153y-595$$

Subtract 153y from both sides.

$$7y+11-34y^{2}-153y=-595$$

Combine 7y and -153y to get -146y.

$$-146y+11-34y^{2}=-595$$

Subtract 11 from both sides.

$$-146y-34y^{2}=-595-11$$

Subtract 11 from -595 to get -606.

$$-146y-34y^{2}=-606$$

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.

$$-34y^{2}-146y=-606$$

Divide both sides by -34.

$$\frac{-34y^{2}-146y}{-34}=-\frac{606}{-34}$$

Dividing by -34 undoes the multiplication by -34.

$$y^{2}+\left(-\frac{146}{-34}\right)y=-\frac{606}{-34}$$

Reduce the fraction \frac{-146}{-34} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{73}{17}y=-\frac{606}{-34}$$

Reduce the fraction \frac{-606}{-34} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{73}{17}y=\frac{303}{17}$$

Divide \frac{73}{17}, the coefficient of the x term, by 2 to get \frac{73}{34}. Then add the square of \frac{73}{34} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{73}{17}y+\left(\frac{73}{34}\right)^{2}=\frac{303}{17}+\left(\frac{73}{34}\right)^{2}$$

Square \frac{73}{34} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{73}{17}y+\frac{5329}{1156}=\frac{303}{17}+\frac{5329}{1156}$$

Add \frac{303}{17} to \frac{5329}{1156} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{73}{17}y+\frac{5329}{1156}=\frac{25933}{1156}$$

Factor y^{2}+\frac{73}{17}y+\frac{5329}{1156}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{73}{34}\right)^{2}=\frac{25933}{1156}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{73}{34}\right)^{2}}=\sqrt{\frac{25933}{1156}}$$

Simplify.

$$y+\frac{73}{34}=\frac{\sqrt{25933}}{34} y+\frac{73}{34}=-\frac{\sqrt{25933}}{34}$$

Subtract \frac{73}{34} from both sides of the equation.

$$y=\frac{\sqrt{25933}-73}{34} y=\frac{-\sqrt{25933}-73}{34}$$