Solve 0=1/2x^2-7x+45/2

$$0 = \frac { 1 } { 2 } x ^ { 2 } - 7 x + \frac { 45 } { 2 }$$

Solve for x

$$x=5$$

$$x=9$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}x^{2}-7x+\frac{45}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -7 for b, and \frac{45}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times \frac{1}{2}\times \frac{45}{2}}}{2\times \frac{1}{2}}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times \frac{1}{2}\times \frac{45}{2}}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\left(-7\right)±\sqrt{49-2\times \frac{45}{2}}}{2\times \frac{1}{2}}$$

Multiply -2 times \frac{45}{2}.

$$x=\frac{-\left(-7\right)±\sqrt{49-45}}{2\times \frac{1}{2}}$$

Add 49 to -45.

$$x=\frac{-\left(-7\right)±\sqrt{4}}{2\times \frac{1}{2}}$$

Take the square root of 4.

$$x=\frac{-\left(-7\right)±2}{2\times \frac{1}{2}}$$

The opposite of -7 is 7.

$$x=\frac{7±2}{2\times \frac{1}{2}}$$

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

$$x=\frac{7±2}{1}$$

Now solve the equation x=\frac{7±2}{1} when ± is plus. Add 7 to 2.

$$x=\frac{9}{1}$$

Divide 9 by 1.

$$x=9$$

Now solve the equation x=\frac{7±2}{1} when ± is minus. Subtract 2 from 7.

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

Divide 5 by 1.

$$x=5$$

The equation is now solved.

$$x=9 x=5$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}x^{2}-7x+\frac{45}{2}=0$$

Subtract \frac{45}{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{2}x^{2}-7x=-\frac{45}{2}$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-7x}{\frac{1}{2}}=-\frac{\frac{45}{2}}{\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{7}{\frac{1}{2}}\right)x=-\frac{\frac{45}{2}}{\frac{1}{2}}$$

Divide -7 by \frac{1}{2} by multiplying -7 by the reciprocal of \frac{1}{2}.

$$x^{2}-14x=-\frac{\frac{45}{2}}{\frac{1}{2}}$$

Divide -\frac{45}{2} by \frac{1}{2} by multiplying -\frac{45}{2} by the reciprocal of \frac{1}{2}.

$$x^{2}-14x=-45$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-45+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-45+49$$

Add -45 to 49.

$$x^{2}-14x+49=4$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-7=2 x-7=-2$$

Add 7 to both sides of the equation.

$$x=9 x=5$$