Solve x^2/9+-x^2+2x/8=1

$$\frac{ { x }^{ 2 } }{ 9 } + \frac{ - { x }^{ 2 } +2x }{ 8 } =1$$

Solve for x

$$x=6$$

$$x=12$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 72, the least common multiple of 9,8.

$$8x^{2}+9\left(-x^{2}+2x\right)=72$$

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

$$8x^{2}+9\left(-x^{2}\right)+18x=72$$

Subtract 72 from both sides.

$$8x^{2}+9\left(-x^{2}\right)+18x-72=0$$

Multiply 9 and -1 to get -9.

$$8x^{2}-9x^{2}+18x-72=0$$

Combine 8x^{2} and -9x^{2} to get -x^{2}.

$$-x^{2}+18x-72=0$$

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

$$a+b=18 ab=-\left(-72\right)=72$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 18.

$$a=12 b=6$$

Rewrite -x^{2}+18x-72 as \left(-x^{2}+12x\right)+\left(6x-72\right).

$$\left(-x^{2}+12x\right)+\left(6x-72\right)$$

Factor out -x in the first and 6 in the second group.

$$-x\left(x-12\right)+6\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x+6\right)$$

To find equation solutions, solve x-12=0 and -x+6=0.

$$x=12 x=6$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 72, the least common multiple of 9,8.

$$8x^{2}+9\left(-x^{2}+2x\right)=72$$

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

$$8x^{2}+9\left(-x^{2}\right)+18x=72$$

Subtract 72 from both sides.

$$8x^{2}+9\left(-x^{2}\right)+18x-72=0$$

Multiply 9 and -1 to get -9.

$$8x^{2}-9x^{2}+18x-72=0$$

Combine 8x^{2} and -9x^{2} to get -x^{2}.

$$-x^{2}+18x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 18 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-1\right)\left(-72\right)}}{2\left(-1\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-1\right)\left(-72\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-18±\sqrt{324+4\left(-72\right)}}{2\left(-1\right)}$$

Multiply 4 times -72.

$$x=\frac{-18±\sqrt{324-288}}{2\left(-1\right)}$$

Add 324 to -288.

$$x=\frac{-18±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$x=\frac{-18±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-18±6}{-2}$$

Now solve the equation x=\frac{-18±6}{-2} when ± is plus. Add -18 to 6.

$$x=-\frac{12}{-2}$$

Divide -12 by -2.

$$x=6$$

Now solve the equation x=\frac{-18±6}{-2} when ± is minus. Subtract 6 from -18.

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

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=6 x=12$$

Steps for Completing the Square

Multiply both sides of the equation by 72, the least common multiple of 9,8.

$$8x^{2}+9\left(-x^{2}+2x\right)=72$$

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

$$8x^{2}+9\left(-x^{2}\right)+18x=72$$

Multiply 9 and -1 to get -9.

$$8x^{2}-9x^{2}+18x=72$$

Combine 8x^{2} and -9x^{2} to get -x^{2}.

$$-x^{2}+18x=72$$

Divide both sides by -1.

$$\frac{-x^{2}+18x}{-1}=\frac{72}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{18}{-1}x=\frac{72}{-1}$$

Divide 18 by -1.

$$x^{2}-18x=\frac{72}{-1}$$

Divide 72 by -1.

$$x^{2}-18x=-72$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-72+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-72+81$$

Add -72 to 81.

$$x^{2}-18x+81=9$$

Factor x^{2}-18x+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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-9=3 x-9=-3$$

Add 9 to both sides of the equation.

$$x=12 x=6$$