Solve 1/2left(3+-x^2+2x+3right)x+1/2left(-x^2+2x+3right)left(3-xright)-1/23*3=0

$$\frac{ 1 }{ 2 } \left( 3+- { x }^{ 2 } +2x+3 \right) x+ \frac{ 1 }{ 2 } \left( - { x }^{ 2 } +2x+3 \right) \left( 3-x \right) - \frac{ 1 }{ 2 } 3 \cdot 3=0$$

Solve for x

$$x=3$$

$$x=0$$

Steps Using Factoring

Add 3 and 3 to get 6.

$$\frac{1}{2}\left(6-x^{2}+2x\right)x+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2} by 6-x^{2}+2x.

$$\left(3+\frac{1}{2}\left(-x^{2}\right)+x\right)x+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply 3+\frac{1}{2}\left(-x^{2}\right)+x by x.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2} by -x^{2}+2x+3.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\left(\frac{1}{2}\left(-x^{2}\right)+x+\frac{3}{2}\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2}\left(-x^{2}\right)+x+\frac{3}{2} by 3-x and combine like terms.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)-\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{2}x+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Combine 3x and \frac{3}{2}x to get \frac{9}{2}x.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{3}{2}\times 3=0$$

Multiply \frac{3}{2} and 3 to get \frac{9}{2}.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{9}{2}=0$$

Subtract \frac{9}{2} from \frac{9}{2} to get 0.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{9}{2}x+\frac{1}{2}\left(-1\right)x^{3}+\frac{3}{2}\left(-1\right)x^{2}+\frac{1}{2}x^{3}=0$$

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{9}{2}x-\frac{1}{2}x^{3}+\frac{3}{2}\left(-1\right)x^{2}+\frac{1}{2}x^{3}=0$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$\frac{9}{2}x-\frac{1}{2}x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x^{3}=0$$

Combine -\frac{1}{2}x^{3} and \frac{1}{2}x^{3} to get 0.

$$\frac{9}{2}x-\frac{3}{2}x^{2}=0$$

Factor out x.

$$x\left(\frac{9}{2}-\frac{3}{2}x\right)=0$$

To find equation solutions, solve x=0 and \frac{9-3x}{2}=0.

$$x=0 x=3$$

Steps Using the Quadratic Formula

Add 3 and 3 to get 6.

$$\frac{1}{2}\left(6-x^{2}+2x\right)x+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2} by 6-x^{2}+2x.

$$\left(3+\frac{1}{2}\left(-x^{2}\right)+x\right)x+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply 3+\frac{1}{2}\left(-x^{2}\right)+x by x.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2} by -x^{2}+2x+3.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\left(\frac{1}{2}\left(-x^{2}\right)+x+\frac{3}{2}\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2}\left(-x^{2}\right)+x+\frac{3}{2} by 3-x and combine like terms.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)-\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{2}x+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Combine 3x and \frac{3}{2}x to get \frac{9}{2}x.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{3}{2}\times 3=0$$

Multiply \frac{3}{2} and 3 to get \frac{9}{2}.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{9}{2}=0$$

Subtract \frac{9}{2} from \frac{9}{2} to get 0.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{9}{2}x+\frac{1}{2}\left(-1\right)x^{3}+\frac{3}{2}\left(-1\right)x^{2}+\frac{1}{2}x^{3}=0$$

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{9}{2}x-\frac{1}{2}x^{3}+\frac{3}{2}\left(-1\right)x^{2}+\frac{1}{2}x^{3}=0$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$\frac{9}{2}x-\frac{1}{2}x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x^{3}=0$$

Combine -\frac{1}{2}x^{3} and \frac{1}{2}x^{3} to get 0.

$$\frac{9}{2}x-\frac{3}{2}x^{2}=0$$

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.

$$-\frac{3}{2}x^{2}+\frac{9}{2}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{3}{2} for a, \frac{9}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}}}{2\left(-\frac{3}{2}\right)}$$

Take the square root of \left(\frac{9}{2}\right)^{2}.

$$x=\frac{-\frac{9}{2}±\frac{9}{2}}{2\left(-\frac{3}{2}\right)}$$

Multiply 2 times -\frac{3}{2}.

$$x=\frac{-\frac{9}{2}±\frac{9}{2}}{-3}$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{9}{2}}{-3} when ± is plus. Add -\frac{9}{2} to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-3}$$

Divide 0 by -3.

$$x=0$$

Now solve the equation x=\frac{-\frac{9}{2}±\frac{9}{2}}{-3} when ± is minus. Subtract \frac{9}{2} from -\frac{9}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{9}{-3}$$

Divide -9 by -3.

$$x=3$$

The equation is now solved.

$$x=0 x=3$$

Steps for Completing the Square

Add 3 and 3 to get 6.

$$\frac{1}{2}\left(6-x^{2}+2x\right)x+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2} by 6-x^{2}+2x.

$$\left(3+\frac{1}{2}\left(-x^{2}\right)+x\right)x+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply 3+\frac{1}{2}\left(-x^{2}\right)+x by x.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{1}{2}\left(-x^{2}+2x+3\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2} by -x^{2}+2x+3.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\left(\frac{1}{2}\left(-x^{2}\right)+x+\frac{3}{2}\right)\left(3-x\right)-\frac{1}{2}\times 3\times 3=0$$

Use the distributive property to multiply \frac{1}{2}\left(-x^{2}\right)+x+\frac{3}{2} by 3-x and combine like terms.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)-\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{2}x+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$3x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{3}{2}x-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Combine 3x and \frac{3}{2}x to get \frac{9}{2}x.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+x^{2}+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}-x^{2}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{1}{2}\times 3\times 3=0$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{3}{2}\times 3=0$$

Multiply \frac{3}{2} and 3 to get \frac{9}{2}.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}+\frac{9}{2}-\frac{9}{2}=0$$

Subtract \frac{9}{2} from \frac{9}{2} to get 0.

$$\frac{9}{2}x+\frac{1}{2}\left(-x^{2}\right)x+\frac{3}{2}\left(-x^{2}\right)+\frac{1}{2}x^{3}=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{9}{2}x+\frac{1}{2}\left(-1\right)x^{3}+\frac{3}{2}\left(-1\right)x^{2}+\frac{1}{2}x^{3}=0$$

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{9}{2}x-\frac{1}{2}x^{3}+\frac{3}{2}\left(-1\right)x^{2}+\frac{1}{2}x^{3}=0$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$\frac{9}{2}x-\frac{1}{2}x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x^{3}=0$$

Combine -\frac{1}{2}x^{3} and \frac{1}{2}x^{3} to get 0.

$$\frac{9}{2}x-\frac{3}{2}x^{2}=0$$

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.

$$-\frac{3}{2}x^{2}+\frac{9}{2}x=0$$

Divide both sides of the equation by -\frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{3}{2}x^{2}+\frac{9}{2}x}{-\frac{3}{2}}=\frac{0}{-\frac{3}{2}}$$

Dividing by -\frac{3}{2} undoes the multiplication by -\frac{3}{2}.

$$x^{2}+\frac{\frac{9}{2}}{-\frac{3}{2}}x=\frac{0}{-\frac{3}{2}}$$

Divide \frac{9}{2} by -\frac{3}{2} by multiplying \frac{9}{2} by the reciprocal of -\frac{3}{2}.

$$x^{2}-3x=\frac{0}{-\frac{3}{2}}$$

Divide 0 by -\frac{3}{2} by multiplying 0 by the reciprocal of -\frac{3}{2}.

$$x^{2}-3x=0$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{9}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{3}{2} x-\frac{3}{2}=-\frac{3}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=3 x=0$$