Solve 8x+11+7/x=21+65x/7

$$8 x + 11 + \frac { 7 } { x } = \frac { 21 + 65 x } { 7 }$$

Solve for x

$$x=-\frac{7}{9}\approx -0.777777778$$

$$x=7$$

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 7x, the least common multiple of x,7.

$$8x\times 7x+7x\times 11+7\times 7=x\left(21+65x\right)$$

Multiply 8 and 7 to get 56.

$$56xx+7x\times 11+7\times 7=x\left(21+65x\right)$$

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

$$56x^{2}+7x\times 11+7\times 7=x\left(21+65x\right)$$

Multiply 7 and 11 to get 77.

$$56x^{2}+77x+7\times 7=x\left(21+65x\right)$$

Multiply 7 and 7 to get 49.

$$56x^{2}+77x+49=x\left(21+65x\right)$$

Use the distributive property to multiply x by 21+65x.

$$56x^{2}+77x+49=21x+65x^{2}$$

Subtract 21x from both sides.

$$56x^{2}+77x+49-21x=65x^{2}$$

Combine 77x and -21x to get 56x.

$$56x^{2}+56x+49=65x^{2}$$

Subtract 65x^{2} from both sides.

$$56x^{2}+56x+49-65x^{2}=0$$

Combine 56x^{2} and -65x^{2} to get -9x^{2}.

$$-9x^{2}+56x+49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx+49. To find a and b, set up a system to be solved.

$$a+b=56 ab=-9\times 49=-441$$

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 -441.

$$-1,441 -3,147 -7,63 -9,49 -21,21$$

Calculate the sum for each pair.

$$-1+441=440 -3+147=144 -7+63=56 -9+49=40 -21+21=0$$

The solution is the pair that gives sum 56.

$$a=63 b=-7$$

Rewrite -9x^{2}+56x+49 as \left(-9x^{2}+63x\right)+\left(-7x+49\right).

$$\left(-9x^{2}+63x\right)+\left(-7x+49\right)$$

Factor out 9x in the first and 7 in the second group.

$$9x\left(-x+7\right)+7\left(-x+7\right)$$

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

$$\left(-x+7\right)\left(9x+7\right)$$

To find equation solutions, solve -x+7=0 and 9x+7=0.

$$x=7 x=-\frac{7}{9}$$

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 7x, the least common multiple of x,7.

$$8x\times 7x+7x\times 11+7\times 7=x\left(21+65x\right)$$

Multiply 8 and 7 to get 56.

$$56xx+7x\times 11+7\times 7=x\left(21+65x\right)$$

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

$$56x^{2}+7x\times 11+7\times 7=x\left(21+65x\right)$$

Multiply 7 and 11 to get 77.

$$56x^{2}+77x+7\times 7=x\left(21+65x\right)$$

Multiply 7 and 7 to get 49.

$$56x^{2}+77x+49=x\left(21+65x\right)$$

Use the distributive property to multiply x by 21+65x.

$$56x^{2}+77x+49=21x+65x^{2}$$

Subtract 21x from both sides.

$$56x^{2}+77x+49-21x=65x^{2}$$

Combine 77x and -21x to get 56x.

$$56x^{2}+56x+49=65x^{2}$$

Subtract 65x^{2} from both sides.

$$56x^{2}+56x+49-65x^{2}=0$$

Combine 56x^{2} and -65x^{2} to get -9x^{2}.

$$-9x^{2}+56x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 56 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\left(-9\right)\times 49}}{2\left(-9\right)}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\left(-9\right)\times 49}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-56±\sqrt{3136+36\times 49}}{2\left(-9\right)}$$

Multiply 36 times 49.

$$x=\frac{-56±\sqrt{3136+1764}}{2\left(-9\right)}$$

Add 3136 to 1764.

$$x=\frac{-56±\sqrt{4900}}{2\left(-9\right)}$$

Take the square root of 4900.

$$x=\frac{-56±70}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-56±70}{-18}$$

Now solve the equation x=\frac{-56±70}{-18} when ± is plus. Add -56 to 70.

$$x=\frac{14}{-18}$$

Reduce the fraction \frac{14}{-18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{9}$$

Now solve the equation x=\frac{-56±70}{-18} when ± is minus. Subtract 70 from -56.

$$x=-\frac{126}{-18}$$

Divide -126 by -18.

$$x=7$$

The equation is now solved.

$$x=-\frac{7}{9} x=7$$

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 7x, the least common multiple of x,7.

$$8x\times 7x+7x\times 11+7\times 7=x\left(21+65x\right)$$

Multiply 8 and 7 to get 56.

$$56xx+7x\times 11+7\times 7=x\left(21+65x\right)$$

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

$$56x^{2}+7x\times 11+7\times 7=x\left(21+65x\right)$$

Multiply 7 and 11 to get 77.

$$56x^{2}+77x+7\times 7=x\left(21+65x\right)$$

Multiply 7 and 7 to get 49.

$$56x^{2}+77x+49=x\left(21+65x\right)$$

Use the distributive property to multiply x by 21+65x.

$$56x^{2}+77x+49=21x+65x^{2}$$

Subtract 21x from both sides.

$$56x^{2}+77x+49-21x=65x^{2}$$

Combine 77x and -21x to get 56x.

$$56x^{2}+56x+49=65x^{2}$$

Subtract 65x^{2} from both sides.

$$56x^{2}+56x+49-65x^{2}=0$$

Combine 56x^{2} and -65x^{2} to get -9x^{2}.

$$-9x^{2}+56x+49=0$$

Subtract 49 from both sides. Anything subtracted from zero gives its negation.

$$-9x^{2}+56x=-49$$

Divide both sides by -9.

$$\frac{-9x^{2}+56x}{-9}=-\frac{49}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{56}{-9}x=-\frac{49}{-9}$$

Divide 56 by -9.

$$x^{2}-\frac{56}{9}x=-\frac{49}{-9}$$

Divide -49 by -9.

$$x^{2}-\frac{56}{9}x=\frac{49}{9}$$

Divide -\frac{56}{9}, the coefficient of the x term, by 2 to get -\frac{28}{9}. Then add the square of -\frac{28}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{56}{9}x+\left(-\frac{28}{9}\right)^{2}=\frac{49}{9}+\left(-\frac{28}{9}\right)^{2}$$

Square -\frac{28}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{56}{9}x+\frac{784}{81}=\frac{49}{9}+\frac{784}{81}$$

Add \frac{49}{9} to \frac{784}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{56}{9}x+\frac{784}{81}=\frac{1225}{81}$$

Factor x^{2}-\frac{56}{9}x+\frac{784}{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}.

$$\left(x-\frac{28}{9}\right)^{2}=\frac{1225}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{28}{9}\right)^{2}}=\sqrt{\frac{1225}{81}}$$

Simplify.

$$x-\frac{28}{9}=\frac{35}{9} x-\frac{28}{9}=-\frac{35}{9}$$

Add \frac{28}{9} to both sides of the equation.

$$x=7 x=-\frac{7}{9}$$