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

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

Solve for x

$$x = \frac{\sqrt{233} - 3}{8} \approx 1.53304219$$

$$x=\frac{-\sqrt{233}-3}{8}\approx -2.28304219$$

Steps Using the Quadratic Formula

Subtract \frac{3}{4}x from both sides.

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

Add \frac{1}{2} to both sides.

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

Add 3 and \frac{1}{2} to get \frac{7}{2}.

$$\frac{7}{2}-x^{2}-\frac{3}{4}x=0$$

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.

$$-x^{2}-\frac{3}{4}x+\frac{7}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{3}{4} for b, and \frac{7}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\left(-\frac{3}{4}\right)^{2}-4\left(-1\right)\times \frac{7}{2}}}{2\left(-1\right)}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-4\left(-1\right)\times \frac{7}{2}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+4\times \frac{7}{2}}}{2\left(-1\right)}$$

Multiply 4 times \frac{7}{2}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+14}}{2\left(-1\right)}$$

Add \frac{9}{16} to 14.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{233}{16}}}{2\left(-1\right)}$$

Take the square root of \frac{233}{16}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\frac{\sqrt{233}}{4}}{2\left(-1\right)}$$

The opposite of -\frac{3}{4} is \frac{3}{4}.

$$x=\frac{\frac{3}{4}±\frac{\sqrt{233}}{4}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{3}{4}±\frac{\sqrt{233}}{4}}{-2}$$

Now solve the equation x=\frac{\frac{3}{4}±\frac{\sqrt{233}}{4}}{-2} when ± is plus. Add \frac{3}{4} to \frac{\sqrt{233}}{4}.

$$x=\frac{\sqrt{233}+3}{-2\times 4}$$

Divide \frac{3+\sqrt{233}}{4} by -2.

$$x=\frac{-\sqrt{233}-3}{8}$$

Now solve the equation x=\frac{\frac{3}{4}±\frac{\sqrt{233}}{4}}{-2} when ± is minus. Subtract \frac{\sqrt{233}}{4} from \frac{3}{4}.

$$x=\frac{3-\sqrt{233}}{-2\times 4}$$

Divide \frac{3-\sqrt{233}}{4} by -2.

$$x=\frac{\sqrt{233}-3}{8}$$

The equation is now solved.

$$x=\frac{-\sqrt{233}-3}{8} x=\frac{\sqrt{233}-3}{8}$$

Steps for Completing the Square

Subtract \frac{3}{4}x from both sides.

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

Subtract 3 from both sides.

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

Subtract 3 from -\frac{1}{2} to get -\frac{7}{2}.

$$-x^{2}-\frac{3}{4}x=-\frac{7}{2}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{3}{4}x}{-1}=-\frac{\frac{7}{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{3}{4}}{-1}\right)x=-\frac{\frac{7}{2}}{-1}$$

Divide -\frac{3}{4} by -1.

$$x^{2}+\frac{3}{4}x=-\frac{\frac{7}{2}}{-1}$$

Divide -\frac{7}{2} by -1.

$$x^{2}+\frac{3}{4}x=\frac{7}{2}$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=\frac{7}{2}+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{7}{2}+\frac{9}{64}$$

Add \frac{7}{2} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{233}{64}$$

Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{8}\right)^{2}=\frac{233}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{\frac{233}{64}}$$

Simplify.

$$x+\frac{3}{8}=\frac{\sqrt{233}}{8} x+\frac{3}{8}=-\frac{\sqrt{233}}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$x=\frac{\sqrt{233}-3}{8} x=\frac{-\sqrt{233}-3}{8}$$