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

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

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{55}i}{14}\approx -0.071428571+0.529728463i$$

$$x=\frac{-\sqrt{55}i-1}{14}\approx -0.071428571-0.529728463i$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

Add 1 to both sides.

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

Add -2 and 1 to get -1.

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

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.

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

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

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

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

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

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

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}+14\left(-1\right)}}{2\left(-\frac{7}{2}\right)}$$

Multiply 14 times -1.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-14}}{2\left(-\frac{7}{2}\right)}$$

Add \frac{1}{4} to -14.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{-\frac{55}{4}}}{2\left(-\frac{7}{2}\right)}$$

Take the square root of -\frac{55}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{\sqrt{55}i}{2}}{2\left(-\frac{7}{2}\right)}$$

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

$$x=\frac{\frac{1}{2}±\frac{\sqrt{55}i}{2}}{2\left(-\frac{7}{2}\right)}$$

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

$$x=\frac{\frac{1}{2}±\frac{\sqrt{55}i}{2}}{-7}$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{\sqrt{55}i}{2}}{-7} when ± is plus. Add \frac{1}{2} to \frac{i\sqrt{55}}{2}.

$$x=\frac{1+\sqrt{55}i}{-7\times 2}$$

Divide \frac{1+i\sqrt{55}}{2} by -7.

$$x=\frac{-\sqrt{55}i-1}{14}$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{\sqrt{55}i}{2}}{-7} when ± is minus. Subtract \frac{i\sqrt{55}}{2} from \frac{1}{2}.

$$x=\frac{-\sqrt{55}i+1}{-7\times 2}$$

Divide \frac{1-i\sqrt{55}}{2} by -7.

$$x=\frac{-1+\sqrt{55}i}{14}$$

The equation is now solved.

$$x=\frac{-\sqrt{55}i-1}{14} x=\frac{-1+\sqrt{55}i}{14}$$

Steps for Completing the Square

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

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

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

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

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

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

Add 2 to both sides.

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

Add -1 and 2 to get 1.

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

Divide both sides of the equation by -\frac{7}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

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

Dividing by -\frac{7}{2} undoes the multiplication by -\frac{7}{2}.

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

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

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

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

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

Divide \frac{1}{7}, the coefficient of the x term, by 2 to get \frac{1}{14}. Then add the square of \frac{1}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{7}x+\left(\frac{1}{14}\right)^{2}=-\frac{2}{7}+\left(\frac{1}{14}\right)^{2}$$

Square \frac{1}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=-\frac{2}{7}+\frac{1}{196}$$

Add -\frac{2}{7} to \frac{1}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=-\frac{55}{196}$$

Factor x^{2}+\frac{1}{7}x+\frac{1}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{14}\right)^{2}=-\frac{55}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{14}\right)^{2}}=\sqrt{-\frac{55}{196}}$$

Simplify.

$$x+\frac{1}{14}=\frac{\sqrt{55}i}{14} x+\frac{1}{14}=-\frac{\sqrt{55}i}{14}$$

Subtract \frac{1}{14} from both sides of the equation.

$$x=\frac{-1+\sqrt{55}i}{14} x=\frac{-\sqrt{55}i-1}{14}$$