Solve (6-x)^2/5+x^2=16/9

$$\frac { ( 6 - x ) ^ { 2 } } { 5 + x ^ { 2 } } = \frac { 16 } { 9 }$$

Solve for x

$$x = -\frac{122}{7} = -17\frac{3}{7} \approx -17.428571429$$

$$x=2$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 9\left(x^{2}+5\right), the least common multiple of 5+x^{2},9.

$$9\left(6-x\right)^{2}=16\left(x^{2}+5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

$$9\left(36-12x+x^{2}\right)=16\left(x^{2}+5\right)$$

Use the distributive property to multiply 9 by 36-12x+x^{2}.

$$324-108x+9x^{2}=16\left(x^{2}+5\right)$$

Use the distributive property to multiply 16 by x^{2}+5.

$$324-108x+9x^{2}=16x^{2}+80$$

Subtract 16x^{2} from both sides.

$$324-108x+9x^{2}-16x^{2}=80$$

Combine 9x^{2} and -16x^{2} to get -7x^{2}.

$$324-108x-7x^{2}=80$$

Subtract 80 from both sides.

$$324-108x-7x^{2}-80=0$$

Subtract 80 from 324 to get 244.

$$244-108x-7x^{2}=0$$

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

$$-7x^{2}-108x+244=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx+244. To find a and b, set up a system to be solved.

$$a+b=-108 ab=-7\times 244=-1708$$

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 -1708.

$$1,-1708 2,-854 4,-427 7,-244 14,-122 28,-61$$

Calculate the sum for each pair.

$$1-1708=-1707 2-854=-852 4-427=-423 7-244=-237 14-122=-108 28-61=-33$$

The solution is the pair that gives sum -108.

$$a=14 b=-122$$

Rewrite -7x^{2}-108x+244 as \left(-7x^{2}+14x\right)+\left(-122x+244\right).

$$\left(-7x^{2}+14x\right)+\left(-122x+244\right)$$

Factor out 7x in the first and 122 in the second group.

$$7x\left(-x+2\right)+122\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(7x+122\right)$$

To find equation solutions, solve -x+2=0 and 7x+122=0.

$$x=2 x=-\frac{122}{7}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 9\left(x^{2}+5\right), the least common multiple of 5+x^{2},9.

$$9\left(6-x\right)^{2}=16\left(x^{2}+5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

$$9\left(36-12x+x^{2}\right)=16\left(x^{2}+5\right)$$

Use the distributive property to multiply 9 by 36-12x+x^{2}.

$$324-108x+9x^{2}=16\left(x^{2}+5\right)$$

Use the distributive property to multiply 16 by x^{2}+5.

$$324-108x+9x^{2}=16x^{2}+80$$

Subtract 16x^{2} from both sides.

$$324-108x+9x^{2}-16x^{2}=80$$

Combine 9x^{2} and -16x^{2} to get -7x^{2}.

$$324-108x-7x^{2}=80$$

Subtract 80 from both sides.

$$324-108x-7x^{2}-80=0$$

Subtract 80 from 324 to get 244.

$$244-108x-7x^{2}=0$$

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.

$$-7x^{2}-108x+244=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -108 for b, and 244 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-108\right)±\sqrt{\left(-108\right)^{2}-4\left(-7\right)\times 244}}{2\left(-7\right)}$$

Square -108.

$$x=\frac{-\left(-108\right)±\sqrt{11664-4\left(-7\right)\times 244}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-108\right)±\sqrt{11664+28\times 244}}{2\left(-7\right)}$$

Multiply 28 times 244.

$$x=\frac{-\left(-108\right)±\sqrt{11664+6832}}{2\left(-7\right)}$$

Add 11664 to 6832.

$$x=\frac{-\left(-108\right)±\sqrt{18496}}{2\left(-7\right)}$$

Take the square root of 18496.

$$x=\frac{-\left(-108\right)±136}{2\left(-7\right)}$$

The opposite of -108 is 108.

$$x=\frac{108±136}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{108±136}{-14}$$

Now solve the equation x=\frac{108±136}{-14} when ± is plus. Add 108 to 136.

$$x=\frac{244}{-14}$$

Reduce the fraction \frac{244}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{122}{7}$$

Now solve the equation x=\frac{108±136}{-14} when ± is minus. Subtract 136 from 108.

$$x=-\frac{28}{-14}$$

Divide -28 by -14.

$$x=2$$

The equation is now solved.

$$x=-\frac{122}{7} x=2$$

Steps for Completing the Square

Multiply both sides of the equation by 9\left(x^{2}+5\right), the least common multiple of 5+x^{2},9.

$$9\left(6-x\right)^{2}=16\left(x^{2}+5\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-x\right)^{2}.

$$9\left(36-12x+x^{2}\right)=16\left(x^{2}+5\right)$$

Use the distributive property to multiply 9 by 36-12x+x^{2}.

$$324-108x+9x^{2}=16\left(x^{2}+5\right)$$

Use the distributive property to multiply 16 by x^{2}+5.

$$324-108x+9x^{2}=16x^{2}+80$$

Subtract 16x^{2} from both sides.

$$324-108x+9x^{2}-16x^{2}=80$$

Combine 9x^{2} and -16x^{2} to get -7x^{2}.

$$324-108x-7x^{2}=80$$

Subtract 324 from both sides.

$$-108x-7x^{2}=80-324$$

Subtract 324 from 80 to get -244.

$$-108x-7x^{2}=-244$$

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.

$$-7x^{2}-108x=-244$$

Divide both sides by -7.

$$\frac{-7x^{2}-108x}{-7}=-\frac{244}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\left(-\frac{108}{-7}\right)x=-\frac{244}{-7}$$

Divide -108 by -7.

$$x^{2}+\frac{108}{7}x=-\frac{244}{-7}$$

Divide -244 by -7.

$$x^{2}+\frac{108}{7}x=\frac{244}{7}$$

Divide \frac{108}{7}, the coefficient of the x term, by 2 to get \frac{54}{7}. Then add the square of \frac{54}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{108}{7}x+\left(\frac{54}{7}\right)^{2}=\frac{244}{7}+\left(\frac{54}{7}\right)^{2}$$

Square \frac{54}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{108}{7}x+\frac{2916}{49}=\frac{244}{7}+\frac{2916}{49}$$

Add \frac{244}{7} to \frac{2916}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{108}{7}x+\frac{2916}{49}=\frac{4624}{49}$$

Factor x^{2}+\frac{108}{7}x+\frac{2916}{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}.

$$\left(x+\frac{54}{7}\right)^{2}=\frac{4624}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{54}{7}\right)^{2}}=\sqrt{\frac{4624}{49}}$$

Simplify.

$$x+\frac{54}{7}=\frac{68}{7} x+\frac{54}{7}=-\frac{68}{7}$$

Subtract \frac{54}{7} from both sides of the equation.

$$x=2 x=-\frac{122}{7}$$