Solve 1/2xleft(x-1right)=45

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

Solve for x

$$x=10$$

$$x=-9$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

Subtract 45 from both sides.

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

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

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

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

Multiply -2 times -45.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}+90}}{2\times \frac{1}{2}}$$

Add \frac{1}{4} to 90.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{361}{4}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{361}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{19}{2}}{2\times \frac{1}{2}}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$x=\frac{\frac{1}{2}±\frac{19}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{\frac{1}{2}±\frac{19}{2}}{1}$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{19}{2}}{1} when ± is plus. Add \frac{1}{2} to \frac{19}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{10}{1}$$

Divide 10 by 1.

$$x=10$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{19}{2}}{1} when ± is minus. Subtract \frac{19}{2} from \frac{1}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -9 by 1.

$$x=-9$$

The equation is now solved.

$$x=10 x=-9$$

Steps for Completing the Square

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

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

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

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

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

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

Multiply both sides by 2.

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

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

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

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

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

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

$$x^{2}-x=90$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=90+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=90+\frac{1}{4}$$

Add 90 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{361}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{19}{2} x-\frac{1}{2}=-\frac{19}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=10 x=-9$$