Solve 7x^2-513/7x^2+47x-72=1/7xdiv9+x+9/x+8

$$\frac { 7 x ^ { 2 } - 513 } { 7 x ^ { 2 } + 47 x - 72 } = \frac { 1 } { 7 x \div 9 } + \frac { x + 9 } { x + 8 }$$

Solve for x

$$x=\frac{9}{49}\approx 0.183673469$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -8,\frac{9}{7} since division by zero is not defined. Multiply both sides of the equation by \left(7x-9\right)\left(x+8\right), the least common multiple of 7x^{2}+47x-72,x+8.

$$7x^{2}-513=9x+\frac{423}{7}-\frac{648}{7}x^{-1}+\left(7x-9\right)\left(x+9\right)$$

Use the distributive property to multiply 7x-9 by x+9 and combine like terms.

$$7x^{2}-513=9x+\frac{423}{7}-\frac{648}{7}x^{-1}+7x^{2}+54x-81$$

Combine 9x and 54x to get 63x.

$$7x^{2}-513=63x+\frac{423}{7}-\frac{648}{7}x^{-1}+7x^{2}-81$$

Subtract 81 from \frac{423}{7} to get -\frac{144}{7}.

$$7x^{2}-513=63x-\frac{144}{7}-\frac{648}{7}x^{-1}+7x^{2}$$

Subtract 63x from both sides.

$$7x^{2}-513-63x=-\frac{144}{7}-\frac{648}{7}x^{-1}+7x^{2}$$

Subtract -\frac{144}{7} from both sides.

$$7x^{2}-513-63x-\left(-\frac{144}{7}\right)=-\frac{648}{7}x^{-1}+7x^{2}$$

The opposite of -\frac{144}{7} is \frac{144}{7}.

$$7x^{2}-513-63x+\frac{144}{7}=-\frac{648}{7}x^{-1}+7x^{2}$$

Add \frac{648}{7}x^{-1} to both sides.

$$7x^{2}-513-63x+\frac{144}{7}+\frac{648}{7}x^{-1}=7x^{2}$$

Add -513 and \frac{144}{7} to get -\frac{3447}{7}.

$$7x^{2}-\frac{3447}{7}-63x+\frac{648}{7}x^{-1}=7x^{2}$$

Subtract 7x^{2} from both sides.

$$7x^{2}-\frac{3447}{7}-63x+\frac{648}{7}x^{-1}-7x^{2}=0$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-\frac{3447}{7}-63x+\frac{648}{7}x^{-1}=0$$

Reorder the terms.

$$-63x-\frac{3447}{7}+\frac{648}{7}\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x, the least common multiple of 7,x.

$$-63x\times 7x+7x\left(-\frac{3447}{7}\right)+\frac{648}{7}\times 7\times 1=0$$

Multiply -63 and 7 to get -441.

$$-441xx+7x\left(-\frac{3447}{7}\right)+\frac{648}{7}\times 7\times 1=0$$

Multiply x and x to get x^{2}.

$$-441x^{2}+7x\left(-\frac{3447}{7}\right)+\frac{648}{7}\times 7\times 1=0$$

Multiply 7 and -\frac{3447}{7} to get -3447.

$$-441x^{2}-3447x+\frac{648}{7}\times 7\times 1=0$$

Multiply \frac{648}{7} and 7 to get 648.

$$-441x^{2}-3447x+648\times 1=0$$

Multiply 648 and 1 to get 648.

$$-441x^{2}-3447x+648=0$$

Divide both sides by 9.

$$-49x^{2}-383x+72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -49x^{2}+ax+bx+72. To find a and b, set up a system to be solved.

$$a+b=-383 ab=-49\times 72=-3528$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -3528.

$$1,-3528 2,-1764 3,-1176 4,-882 6,-588 7,-504 8,-441 9,-392 12,-294 14,-252 18,-196 21,-168 24,-147 28,-126 36,-98 42,-84 49,-72 56,-63$$

Calculate the sum for each pair.

$$1-3528=-3527 2-1764=-1762 3-1176=-1173 4-882=-878 6-588=-582 7-504=-497 8-441=-433 9-392=-383 12-294=-282 14-252=-238 18-196=-178 21-168=-147 24-147=-123 28-126=-98 36-98=-62 42-84=-42 49-72=-23 56-63=-7$$

The solution is the pair that gives sum -383.

$$a=9 b=-392$$

Rewrite -49x^{2}-383x+72 as \left(-49x^{2}+9x\right)+\left(-392x+72\right).

$$\left(-49x^{2}+9x\right)+\left(-392x+72\right)$$

Factor out -x in the first and -8 in the second group.

$$-x\left(49x-9\right)-8\left(49x-9\right)$$

Factor out common term 49x-9 by using distributive property.

$$\left(49x-9\right)\left(-x-8\right)$$

To find equation solutions, solve 49x-9=0 and -x-8=0.

$$x=\frac{9}{49} x=-8$$

Variable x cannot be equal to -8.

$$x=\frac{9}{49}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -8,\frac{9}{7} since division by zero is not defined. Multiply both sides of the equation by \left(7x-9\right)\left(x+8\right), the least common multiple of 7x^{2}+47x-72,x+8.

$$7x^{2}-513=9x+\frac{423}{7}-\frac{648}{7}x^{-1}+\left(7x-9\right)\left(x+9\right)$$

Use the distributive property to multiply 7x-9 by x+9 and combine like terms.

$$7x^{2}-513=9x+\frac{423}{7}-\frac{648}{7}x^{-1}+7x^{2}+54x-81$$

Combine 9x and 54x to get 63x.

$$7x^{2}-513=63x+\frac{423}{7}-\frac{648}{7}x^{-1}+7x^{2}-81$$

Subtract 81 from \frac{423}{7} to get -\frac{144}{7}.

$$7x^{2}-513=63x-\frac{144}{7}-\frac{648}{7}x^{-1}+7x^{2}$$

Subtract 63x from both sides.

$$7x^{2}-513-63x=-\frac{144}{7}-\frac{648}{7}x^{-1}+7x^{2}$$

Subtract -\frac{144}{7} from both sides.

$$7x^{2}-513-63x-\left(-\frac{144}{7}\right)=-\frac{648}{7}x^{-1}+7x^{2}$$

The opposite of -\frac{144}{7} is \frac{144}{7}.

$$7x^{2}-513-63x+\frac{144}{7}=-\frac{648}{7}x^{-1}+7x^{2}$$

Add \frac{648}{7}x^{-1} to both sides.

$$7x^{2}-513-63x+\frac{144}{7}+\frac{648}{7}x^{-1}=7x^{2}$$

Add -513 and \frac{144}{7} to get -\frac{3447}{7}.

$$7x^{2}-\frac{3447}{7}-63x+\frac{648}{7}x^{-1}=7x^{2}$$

Subtract 7x^{2} from both sides.

$$7x^{2}-\frac{3447}{7}-63x+\frac{648}{7}x^{-1}-7x^{2}=0$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-\frac{3447}{7}-63x+\frac{648}{7}x^{-1}=0$$

Reorder the terms.

$$-63x-\frac{3447}{7}+\frac{648}{7}\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x, the least common multiple of 7,x.

$$-63x\times 7x+7x\left(-\frac{3447}{7}\right)+\frac{648}{7}\times 7\times 1=0$$

Multiply -63 and 7 to get -441.

$$-441xx+7x\left(-\frac{3447}{7}\right)+\frac{648}{7}\times 7\times 1=0$$

Multiply x and x to get x^{2}.

$$-441x^{2}+7x\left(-\frac{3447}{7}\right)+\frac{648}{7}\times 7\times 1=0$$

Multiply 7 and -\frac{3447}{7} to get -3447.

$$-441x^{2}-3447x+\frac{648}{7}\times 7\times 1=0$$

Multiply \frac{648}{7} and 7 to get 648.

$$-441x^{2}-3447x+648\times 1=0$$

Multiply 648 and 1 to get 648.

$$-441x^{2}-3447x+648=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -441 for a, -3447 for b, and 648 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3447\right)±\sqrt{\left(-3447\right)^{2}-4\left(-441\right)\times 648}}{2\left(-441\right)}$$

Square -3447.

$$x=\frac{-\left(-3447\right)±\sqrt{11881809-4\left(-441\right)\times 648}}{2\left(-441\right)}$$

Multiply -4 times -441.

$$x=\frac{-\left(-3447\right)±\sqrt{11881809+1764\times 648}}{2\left(-441\right)}$$

Multiply 1764 times 648.

$$x=\frac{-\left(-3447\right)±\sqrt{11881809+1143072}}{2\left(-441\right)}$$

Add 11881809 to 1143072.

$$x=\frac{-\left(-3447\right)±\sqrt{13024881}}{2\left(-441\right)}$$

Take the square root of 13024881.

$$x=\frac{-\left(-3447\right)±3609}{2\left(-441\right)}$$

The opposite of -3447 is 3447.

$$x=\frac{3447±3609}{2\left(-441\right)}$$

Multiply 2 times -441.

$$x=\frac{3447±3609}{-882}$$

Now solve the equation x=\frac{3447±3609}{-882} when ± is plus. Add 3447 to 3609.

$$x=\frac{7056}{-882}$$

Divide 7056 by -882.

$$x=-8$$

Now solve the equation x=\frac{3447±3609}{-882} when ± is minus. Subtract 3609 from 3447.

$$x=-\frac{162}{-882}$$

Reduce the fraction \frac{-162}{-882} to lowest terms by extracting and canceling out 18.

$$x=\frac{9}{49}$$

The equation is now solved.

$$x=-8 x=\frac{9}{49}$$

Variable x cannot be equal to -8.

$$x=\frac{9}{49}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -8,\frac{9}{7} since division by zero is not defined. Multiply both sides of the equation by \left(7x-9\right)\left(x+8\right), the least common multiple of 7x^{2}+47x-72,x+8.

$$7x^{2}-513=9x+\frac{423}{7}-\frac{648}{7}x^{-1}+\left(7x-9\right)\left(x+9\right)$$

Use the distributive property to multiply 7x-9 by x+9 and combine like terms.

$$7x^{2}-513=9x+\frac{423}{7}-\frac{648}{7}x^{-1}+7x^{2}+54x-81$$

Combine 9x and 54x to get 63x.

$$7x^{2}-513=63x+\frac{423}{7}-\frac{648}{7}x^{-1}+7x^{2}-81$$

Subtract 81 from \frac{423}{7} to get -\frac{144}{7}.

$$7x^{2}-513=63x-\frac{144}{7}-\frac{648}{7}x^{-1}+7x^{2}$$

Subtract 63x from both sides.

$$7x^{2}-513-63x=-\frac{144}{7}-\frac{648}{7}x^{-1}+7x^{2}$$

Add \frac{648}{7}x^{-1} to both sides.

$$7x^{2}-513-63x+\frac{648}{7}x^{-1}=-\frac{144}{7}+7x^{2}$$

Subtract 7x^{2} from both sides.

$$7x^{2}-513-63x+\frac{648}{7}x^{-1}-7x^{2}=-\frac{144}{7}$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-513-63x+\frac{648}{7}x^{-1}=-\frac{144}{7}$$

Add 513 to both sides.

$$-63x+\frac{648}{7}x^{-1}=-\frac{144}{7}+513$$

Add -\frac{144}{7} and 513 to get \frac{3447}{7}.

$$-63x+\frac{648}{7}x^{-1}=\frac{3447}{7}$$

Reorder the terms.

$$-63x+\frac{648}{7}\times \frac{1}{x}=\frac{3447}{7}$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7x, the least common multiple of 7,x.

$$-63x\times 7x+\frac{648}{7}\times 7\times 1=3447x$$

Multiply -63 and 7 to get -441.

$$-441xx+\frac{648}{7}\times 7\times 1=3447x$$

Multiply x and x to get x^{2}.

$$-441x^{2}+\frac{648}{7}\times 7\times 1=3447x$$

Multiply \frac{648}{7} and 7 to get 648.

$$-441x^{2}+648\times 1=3447x$$

Multiply 648 and 1 to get 648.

$$-441x^{2}+648=3447x$$

Subtract 3447x from both sides.

$$-441x^{2}+648-3447x=0$$

Subtract 648 from both sides. Anything subtracted from zero gives its negation.

$$-441x^{2}-3447x=-648$$

Divide both sides by -441.

$$\frac{-441x^{2}-3447x}{-441}=-\frac{648}{-441}$$

Dividing by -441 undoes the multiplication by -441.

$$x^{2}+\left(-\frac{3447}{-441}\right)x=-\frac{648}{-441}$$

Reduce the fraction \frac{-3447}{-441} to lowest terms by extracting and canceling out 9.

$$x^{2}+\frac{383}{49}x=-\frac{648}{-441}$$

Reduce the fraction \frac{-648}{-441} to lowest terms by extracting and canceling out 9.

$$x^{2}+\frac{383}{49}x=\frac{72}{49}$$

Divide \frac{383}{49}, the coefficient of the x term, by 2 to get \frac{383}{98}. Then add the square of \frac{383}{98} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{383}{49}x+\left(\frac{383}{98}\right)^{2}=\frac{72}{49}+\left(\frac{383}{98}\right)^{2}$$

Square \frac{383}{98} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{383}{49}x+\frac{146689}{9604}=\frac{72}{49}+\frac{146689}{9604}$$

Add \frac{72}{49} to \frac{146689}{9604} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{383}{49}x+\frac{146689}{9604}=\frac{160801}{9604}$$

Factor x^{2}+\frac{383}{49}x+\frac{146689}{9604}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{383}{98}\right)^{2}=\frac{160801}{9604}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{383}{98}\right)^{2}}=\sqrt{\frac{160801}{9604}}$$

Simplify.

$$x+\frac{383}{98}=\frac{401}{98} x+\frac{383}{98}=-\frac{401}{98}$$

Subtract \frac{383}{98} from both sides of the equation.

$$x=\frac{9}{49} x=-8$$

Variable x cannot be equal to -8.

$$x=\frac{9}{49}$$