Solve 1/2x^2-3x+21/2-x-3=0

$$\frac { 1 } { 2 } x ^ { 2 } - 3 x + \frac { 21 } { 2 } - x - 3 = 0$$

Solve for x

$$x=3$$

$$x=5$$

Steps Using the Quadratic Formula

Combine -3x and -x to get -4x.

$$\frac{1}{2}x^{2}-4x+\frac{21}{2}-3=0$$

Subtract 3 from \frac{21}{2} to get \frac{15}{2}.

$$\frac{1}{2}x^{2}-4x+\frac{15}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -4 for b, and \frac{15}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times \frac{1}{2}\times \frac{15}{2}}}{2\times \frac{1}{2}}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times \frac{1}{2}\times \frac{15}{2}}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{-\left(-4\right)±\sqrt{16-2\times \frac{15}{2}}}{2\times \frac{1}{2}}$$

Multiply -2 times \frac{15}{2}.

$$x=\frac{-\left(-4\right)±\sqrt{16-15}}{2\times \frac{1}{2}}$$

Add 16 to -15.

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

Take the square root of 1.

$$x=\frac{-\left(-4\right)±1}{2\times \frac{1}{2}}$$

The opposite of -4 is 4.

$$x=\frac{4±1}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{4±1}{1}$$

Now solve the equation x=\frac{4±1}{1} when ± is plus. Add 4 to 1.

$$x=\frac{5}{1}$$

Divide 5 by 1.

$$x=5$$

Now solve the equation x=\frac{4±1}{1} when ± is minus. Subtract 1 from 4.

$$x=\frac{3}{1}$$

Divide 3 by 1.

$$x=3$$

The equation is now solved.

$$x=5 x=3$$

Steps for Completing the Square

Combine -3x and -x to get -4x.

$$\frac{1}{2}x^{2}-4x+\frac{21}{2}-3=0$$

Subtract 3 from \frac{21}{2} to get \frac{15}{2}.

$$\frac{1}{2}x^{2}-4x+\frac{15}{2}=0$$

Subtract \frac{15}{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{2}x^{2}-4x=-\frac{15}{2}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-4x}{\frac{1}{2}}=-\frac{\frac{15}{2}}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$x^{2}+\left(-\frac{4}{\frac{1}{2}}\right)x=-\frac{\frac{15}{2}}{\frac{1}{2}}$$

Divide -4 by \frac{1}{2} by multiplying -4 by the reciprocal of \frac{1}{2}.

$$x^{2}-8x=-\frac{\frac{15}{2}}{\frac{1}{2}}$$

Divide -\frac{15}{2} by \frac{1}{2} by multiplying -\frac{15}{2} by the reciprocal of \frac{1}{2}.

$$x^{2}-8x=-15$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-15+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-15+16$$

Add -15 to 16.

$$x^{2}-8x+16=1$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=1$$

Take the square root of both sides of the equation.

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

Simplify.

$$x-4=1 x-4=-1$$

Add 4 to both sides of the equation.

$$x=5 x=3$$