Solve 1/2x-3=1/4x^2-x-3

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

Solve for x

$$x=6$$

$$x=0$$

Steps Using Factoring

Subtract \frac{1}{4}x^{2} from both sides.

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

Add x to both sides.

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

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

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

Add 3 to both sides.

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

Add -3 and 3 to get 0.

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

Factor out x.

$$x\left(\frac{3}{2}-\frac{1}{4}x\right)=0$$

To find equation solutions, solve x=0 and \frac{3}{2}-\frac{x}{4}=0.

$$x=0 x=6$$

Steps Using the Quadratic Formula

Subtract \frac{1}{4}x^{2} from both sides.

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

Add x to both sides.

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

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

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

Add 3 to both sides.

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

Add -3 and 3 to get 0.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{4} for a, \frac{3}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{3}{2}±\sqrt{\left(\frac{3}{2}\right)^{2}}}{2\left(-\frac{1}{4}\right)}$$

Take the square root of \left(\frac{3}{2}\right)^{2}.

$$x=\frac{-\frac{3}{2}±\frac{3}{2}}{2\left(-\frac{1}{4}\right)}$$

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

$$x=\frac{-\frac{3}{2}±\frac{3}{2}}{-\frac{1}{2}}$$

Now solve the equation x=\frac{-\frac{3}{2}±\frac{3}{2}}{-\frac{1}{2}} when ± is plus. Add -\frac{3}{2} to \frac{3}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 0 by -\frac{1}{2} by multiplying 0 by the reciprocal of -\frac{1}{2}.

$$x=0$$

Now solve the equation x=\frac{-\frac{3}{2}±\frac{3}{2}}{-\frac{1}{2}} when ± is minus. Subtract \frac{3}{2} from -\frac{3}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3}{-\frac{1}{2}}$$

Divide -3 by -\frac{1}{2} by multiplying -3 by the reciprocal of -\frac{1}{2}.

$$x=6$$

The equation is now solved.

$$x=0 x=6$$

Steps for Completing the Square

Subtract \frac{1}{4}x^{2} from both sides.

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

Add x to both sides.

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

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

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

Add 3 to both sides.

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

Add -3 and 3 to get 0.

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

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.

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

Multiply both sides by -4.

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

Dividing by -\frac{1}{4} undoes the multiplication by -\frac{1}{4}.

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

Divide \frac{3}{2} by -\frac{1}{4} by multiplying \frac{3}{2} by the reciprocal of -\frac{1}{4}.

$$x^{2}-6x=\frac{0}{-\frac{1}{4}}$$

Divide 0 by -\frac{1}{4} by multiplying 0 by the reciprocal of -\frac{1}{4}.

$$x^{2}-6x=0$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -3.

$$x^{2}-6x+9=9$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-3=3 x-3=-3$$

Add 3 to both sides of the equation.

$$x=6 x=0$$