Solve (x-3)^2/2+5x+6/6=(x+2)(x-2)/3+6

$$\frac { ( x - 3 ) ^ { 2 } } { 2 } + \frac { 5 x + 6 } { 6 } = \frac { ( x + 2 ) ( x - 2 ) } { 3 } + 6$$

Solve for x

$$x = \frac{\sqrt{149} + 13}{2} \approx 12.603277808$$

$$x=\frac{13-\sqrt{149}}{2}\approx 0.396722192$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 2,6,3.

$$3\left(x-3\right)^{2}+5x+6=2\left(x+2\right)\left(x-2\right)+36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$3\left(x^{2}-6x+9\right)+5x+6=2\left(x+2\right)\left(x-2\right)+36$$

Use the distributive property to multiply 3 by x^{2}-6x+9.

$$3x^{2}-18x+27+5x+6=2\left(x+2\right)\left(x-2\right)+36$$

Combine -18x and 5x to get -13x.

$$3x^{2}-13x+27+6=2\left(x+2\right)\left(x-2\right)+36$$

Add 27 and 6 to get 33.

$$3x^{2}-13x+33=2\left(x+2\right)\left(x-2\right)+36$$

Use the distributive property to multiply 2 by x+2.

$$3x^{2}-13x+33=\left(2x+4\right)\left(x-2\right)+36$$

Use the distributive property to multiply 2x+4 by x-2 and combine like terms.

$$3x^{2}-13x+33=2x^{2}-8+36$$

Add -8 and 36 to get 28.

$$3x^{2}-13x+33=2x^{2}+28$$

Subtract 2x^{2} from both sides.

$$3x^{2}-13x+33-2x^{2}=28$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}-13x+33=28$$

Subtract 28 from both sides.

$$x^{2}-13x+33-28=0$$

Subtract 28 from 33 to get 5.

$$x^{2}-13x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 5}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 5}}{2}$$

Multiply -4 times 5.

$$x=\frac{-\left(-13\right)±\sqrt{169-20}}{2}$$

Add 169 to -20.

$$x=\frac{-\left(-13\right)±\sqrt{149}}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{149}}{2}$$

Now solve the equation x=\frac{13±\sqrt{149}}{2} when ± is plus. Add 13 to \sqrt{149}.

$$x=\frac{\sqrt{149}+13}{2}$$

Now solve the equation x=\frac{13±\sqrt{149}}{2} when ± is minus. Subtract \sqrt{149} from 13.

$$x=\frac{13-\sqrt{149}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{149}+13}{2} x=\frac{13-\sqrt{149}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 2,6,3.

$$3\left(x-3\right)^{2}+5x+6=2\left(x+2\right)\left(x-2\right)+36$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$3\left(x^{2}-6x+9\right)+5x+6=2\left(x+2\right)\left(x-2\right)+36$$

Use the distributive property to multiply 3 by x^{2}-6x+9.

$$3x^{2}-18x+27+5x+6=2\left(x+2\right)\left(x-2\right)+36$$

Combine -18x and 5x to get -13x.

$$3x^{2}-13x+27+6=2\left(x+2\right)\left(x-2\right)+36$$

Add 27 and 6 to get 33.

$$3x^{2}-13x+33=2\left(x+2\right)\left(x-2\right)+36$$

Use the distributive property to multiply 2 by x+2.

$$3x^{2}-13x+33=\left(2x+4\right)\left(x-2\right)+36$$

Use the distributive property to multiply 2x+4 by x-2 and combine like terms.

$$3x^{2}-13x+33=2x^{2}-8+36$$

Add -8 and 36 to get 28.

$$3x^{2}-13x+33=2x^{2}+28$$

Subtract 2x^{2} from both sides.

$$3x^{2}-13x+33-2x^{2}=28$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}-13x+33=28$$

Subtract 33 from both sides.

$$x^{2}-13x=28-33$$

Subtract 33 from 28 to get -5.

$$x^{2}-13x=-5$$

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

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-5+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=-5+\frac{169}{4}$$

Add -5 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{149}{4}$$

Factor x^{2}-13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{149}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{149}{4}}$$

Simplify.

$$x-\frac{13}{2}=\frac{\sqrt{149}}{2} x-\frac{13}{2}=-\frac{\sqrt{149}}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=\frac{\sqrt{149}+13}{2} x=\frac{13-\sqrt{149}}{2}$$