Solve left(12-2xright)^2+9x^2=100

$${ \left(12-2x \right) }^{ 2 } +9 { x }^{ 2 } =100$$

Solve for x

$$x=2$$

$$x = \frac{22}{13} = 1\frac{9}{13} \approx 1.692307692$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-2x\right)^{2}.

$$144-48x+4x^{2}+9x^{2}=100$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$144-48x+13x^{2}=100$$

Subtract 100 from both sides.

$$144-48x+13x^{2}-100=0$$

Subtract 100 from 144 to get 44.

$$44-48x+13x^{2}=0$$

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

$$13x^{2}-48x+44=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 13x^{2}+ax+bx+44. To find a and b, set up a system to be solved.

$$a+b=-48 ab=13\times 44=572$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 572.

$$-1,-572 -2,-286 -4,-143 -11,-52 -13,-44 -22,-26$$

Calculate the sum for each pair.

$$-1-572=-573 -2-286=-288 -4-143=-147 -11-52=-63 -13-44=-57 -22-26=-48$$

The solution is the pair that gives sum -48.

$$a=-26 b=-22$$

Rewrite 13x^{2}-48x+44 as \left(13x^{2}-26x\right)+\left(-22x+44\right).

$$\left(13x^{2}-26x\right)+\left(-22x+44\right)$$

Factor out 13x in the first and -22 in the second group.

$$13x\left(x-2\right)-22\left(x-2\right)$$

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

$$\left(x-2\right)\left(13x-22\right)$$

To find equation solutions, solve x-2=0 and 13x-22=0.

$$x=2 x=\frac{22}{13}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-2x\right)^{2}.

$$144-48x+4x^{2}+9x^{2}=100$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$144-48x+13x^{2}=100$$

Subtract 100 from both sides.

$$144-48x+13x^{2}-100=0$$

Subtract 100 from 144 to get 44.

$$44-48x+13x^{2}=0$$

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.

$$13x^{2}-48x+44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -48 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 13\times 44}}{2\times 13}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 13\times 44}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-48\right)±\sqrt{2304-52\times 44}}{2\times 13}$$

Multiply -52 times 44.

$$x=\frac{-\left(-48\right)±\sqrt{2304-2288}}{2\times 13}$$

Add 2304 to -2288.

$$x=\frac{-\left(-48\right)±\sqrt{16}}{2\times 13}$$

Take the square root of 16.

$$x=\frac{-\left(-48\right)±4}{2\times 13}$$

The opposite of -48 is 48.

$$x=\frac{48±4}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{48±4}{26}$$

Now solve the equation x=\frac{48±4}{26} when ± is plus. Add 48 to 4.

$$x=\frac{52}{26}$$

Divide 52 by 26.

$$x=2$$

Now solve the equation x=\frac{48±4}{26} when ± is minus. Subtract 4 from 48.

$$x=\frac{44}{26}$$

Reduce the fraction \frac{44}{26} to lowest terms by extracting and canceling out 2.

$$x=\frac{22}{13}$$

The equation is now solved.

$$x=2 x=\frac{22}{13}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-2x\right)^{2}.

$$144-48x+4x^{2}+9x^{2}=100$$

Combine 4x^{2} and 9x^{2} to get 13x^{2}.

$$144-48x+13x^{2}=100$$

Subtract 144 from both sides.

$$-48x+13x^{2}=100-144$$

Subtract 144 from 100 to get -44.

$$-48x+13x^{2}=-44$$

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.

$$13x^{2}-48x=-44$$

Divide both sides by 13.

$$\frac{13x^{2}-48x}{13}=-\frac{44}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}-\frac{48}{13}x=-\frac{44}{13}$$

Divide -\frac{48}{13}, the coefficient of the x term, by 2 to get -\frac{24}{13}. Then add the square of -\frac{24}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{48}{13}x+\left(-\frac{24}{13}\right)^{2}=-\frac{44}{13}+\left(-\frac{24}{13}\right)^{2}$$

Square -\frac{24}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{48}{13}x+\frac{576}{169}=-\frac{44}{13}+\frac{576}{169}$$

Add -\frac{44}{13} to \frac{576}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{48}{13}x+\frac{576}{169}=\frac{4}{169}$$

Factor x^{2}-\frac{48}{13}x+\frac{576}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{24}{13}\right)^{2}=\frac{4}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{24}{13}\right)^{2}}=\sqrt{\frac{4}{169}}$$

Simplify.

$$x-\frac{24}{13}=\frac{2}{13} x-\frac{24}{13}=-\frac{2}{13}$$

Add \frac{24}{13} to both sides of the equation.

$$x=2 x=\frac{22}{13}$$