Solve left(x-1/2right)left(x+2right)+left(x-1/3right)left(x+3right)=xleft(25/6+xright)

$$\left( x- \frac{ 1 }{ 2 } \right) \left( x+2 \right) + \left( x- \frac{ 1 }{ 3 } \right) \left( x+3 \right) =x \left( \frac{ 25 }{ 6 } +x \right)$$

Solve for x

$$x=\sqrt{2}\approx 1.414213562$$

$$x=-\sqrt{2}\approx -1.414213562$$

Steps by Finding Square Root

Use the distributive property to multiply x-\frac{1}{2} by x+2 and combine like terms.

$$x^{2}+\frac{3}{2}x-1+\left(x-\frac{1}{3}\right)\left(x+3\right)=x\left(\frac{25}{6}+x\right)$$

Use the distributive property to multiply x-\frac{1}{3} by x+3 and combine like terms.

$$x^{2}+\frac{3}{2}x-1+x^{2}+\frac{8}{3}x-1=x\left(\frac{25}{6}+x\right)$$

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

$$2x^{2}+\frac{3}{2}x-1+\frac{8}{3}x-1=x\left(\frac{25}{6}+x\right)$$

Combine \frac{3}{2}x and \frac{8}{3}x to get \frac{25}{6}x.

$$2x^{2}+\frac{25}{6}x-1-1=x\left(\frac{25}{6}+x\right)$$

Subtract 1 from -1 to get -2.

$$2x^{2}+\frac{25}{6}x-2=x\left(\frac{25}{6}+x\right)$$

Use the distributive property to multiply x by \frac{25}{6}+x.

$$2x^{2}+\frac{25}{6}x-2=\frac{25}{6}x+x^{2}$$

Subtract \frac{25}{6}x from both sides.

$$2x^{2}+\frac{25}{6}x-2-\frac{25}{6}x=x^{2}$$

Combine \frac{25}{6}x and -\frac{25}{6}x to get 0.

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

Subtract x^{2} from both sides.

$$2x^{2}-2-x^{2}=0$$

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

$$x^{2}-2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$x^{2}=2$$

Take the square root of both sides of the equation.

$$x=\sqrt{2} x=-\sqrt{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-\frac{1}{2} by x+2 and combine like terms.

$$x^{2}+\frac{3}{2}x-1+\left(x-\frac{1}{3}\right)\left(x+3\right)=x\left(\frac{25}{6}+x\right)$$

Use the distributive property to multiply x-\frac{1}{3} by x+3 and combine like terms.

$$x^{2}+\frac{3}{2}x-1+x^{2}+\frac{8}{3}x-1=x\left(\frac{25}{6}+x\right)$$

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

$$2x^{2}+\frac{3}{2}x-1+\frac{8}{3}x-1=x\left(\frac{25}{6}+x\right)$$

Combine \frac{3}{2}x and \frac{8}{3}x to get \frac{25}{6}x.

$$2x^{2}+\frac{25}{6}x-1-1=x\left(\frac{25}{6}+x\right)$$

Subtract 1 from -1 to get -2.

$$2x^{2}+\frac{25}{6}x-2=x\left(\frac{25}{6}+x\right)$$

Use the distributive property to multiply x by \frac{25}{6}+x.

$$2x^{2}+\frac{25}{6}x-2=\frac{25}{6}x+x^{2}$$

Subtract \frac{25}{6}x from both sides.

$$2x^{2}+\frac{25}{6}x-2-\frac{25}{6}x=x^{2}$$

Combine \frac{25}{6}x and -\frac{25}{6}x to get 0.

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

Subtract x^{2} from both sides.

$$2x^{2}-2-x^{2}=0$$

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

$$x^{2}-2=0$$

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{0±\sqrt{8}}{2}$$

Take the square root of 8.

$$x=\frac{0±2\sqrt{2}}{2}$$

Now solve the equation x=\frac{0±2\sqrt{2}}{2} when ± is plus.

$$x=\sqrt{2}$$

Now solve the equation x=\frac{0±2\sqrt{2}}{2} when ± is minus.

$$x=-\sqrt{2}$$

The equation is now solved.

$$x=\sqrt{2} x=-\sqrt{2}$$