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

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

Solve for x

$$x = \frac{\sqrt{17} + 7}{4} \approx 2.780776406$$

$$x=\frac{7-\sqrt{17}}{4}\approx 0.719223594$$

Steps Using the Quadratic Formula

Combine \frac{3}{2}x and \frac{1}{4}x to get \frac{7}{4}x.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x+2-1=2$$

Subtract 1 from 2 to get 1.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x+1=2$$

Subtract 2 from both sides.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x+1-2=0$$

Subtract 2 from 1 to get -1.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x-1=0$$

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

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

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{49}{16}-4\left(-\frac{1}{2}\right)\left(-1\right)}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{49}{16}+2\left(-1\right)}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{49}{16}-2}}{2\left(-\frac{1}{2}\right)}$$

Add \frac{49}{16} to -2.

$$x=\frac{-\frac{7}{4}±\sqrt{\frac{17}{16}}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of \frac{17}{16}.

$$x=\frac{-\frac{7}{4}±\frac{\sqrt{17}}{4}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-\frac{7}{4}±\frac{\sqrt{17}}{4}}{-1}$$

Now solve the equation x=\frac{-\frac{7}{4}±\frac{\sqrt{17}}{4}}{-1} when ± is plus. Add -\frac{7}{4} to \frac{\sqrt{17}}{4}.

$$x=\frac{\sqrt{17}-7}{-4}$$

Divide \frac{-7+\sqrt{17}}{4} by -1.

$$x=\frac{7-\sqrt{17}}{4}$$

Now solve the equation x=\frac{-\frac{7}{4}±\frac{\sqrt{17}}{4}}{-1} when ± is minus. Subtract \frac{\sqrt{17}}{4} from -\frac{7}{4}.

$$x=\frac{-\sqrt{17}-7}{-4}$$

Divide \frac{-7-\sqrt{17}}{4} by -1.

$$x=\frac{\sqrt{17}+7}{4}$$

The equation is now solved.

$$x=\frac{7-\sqrt{17}}{4} x=\frac{\sqrt{17}+7}{4}$$

Steps for Completing the Square

Combine \frac{3}{2}x and \frac{1}{4}x to get \frac{7}{4}x.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x+2-1=2$$

Subtract 1 from 2 to get 1.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x+1=2$$

Subtract 1 from both sides.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x=2-1$$

Subtract 1 from 2 to get 1.

$$-\frac{1}{2}x^{2}+\frac{7}{4}x=1$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}x^{2}+\frac{7}{4}x}{-\frac{1}{2}}=\frac{1}{-\frac{1}{2}}$$

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

$$x^{2}+\frac{\frac{7}{4}}{-\frac{1}{2}}x=\frac{1}{-\frac{1}{2}}$$

Divide \frac{7}{4} by -\frac{1}{2} by multiplying \frac{7}{4} by the reciprocal of -\frac{1}{2}.

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

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

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

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-2+\frac{49}{16}$$

Add -2 to \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{17}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{17}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{\sqrt{17}}{4} x-\frac{7}{4}=-\frac{\sqrt{17}}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=\frac{\sqrt{17}+7}{4} x=\frac{7-\sqrt{17}}{4}$$