Solve -x^2-1=-3x-5.5

$$- x ^ { 2 } - 1 = - 3 x - 5.5$$

Solve for x

$$x = \frac{3 \sqrt{3} + 3}{2} \approx 4.098076211$$

$$x=\frac{3-3\sqrt{3}}{2}\approx -1.098076211$$

Steps Using the Quadratic Formula

Add 3x to both sides.

$$-x^{2}-1+3x=-5.5$$

Add 5.5 to both sides.

$$-x^{2}-1+3x+5.5=0$$

Add -1 and 5.5 to get 4.5.

$$-x^{2}+4.5+3x=0$$

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.

$$-x^{2}+3x+4.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 4.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 4.5}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\times 4.5}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\times 4.5}}{2\left(-1\right)}$$

Multiply 4 times 4.5.

$$x=\frac{-3±\sqrt{9+18}}{2\left(-1\right)}$$

Add 9 to 18.

$$x=\frac{-3±\sqrt{27}}{2\left(-1\right)}$$

Take the square root of 27.

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

Multiply 2 times -1.

$$x=\frac{-3±3\sqrt{3}}{-2}$$

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

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

Divide -3+3\sqrt{3} by -2.

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

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

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

Divide -3-3\sqrt{3} by -2.

$$x=\frac{3\sqrt{3}+3}{2}$$

The equation is now solved.

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

Steps for Completing the Square

Add 3x to both sides.

$$-x^{2}-1+3x=-5.5$$

Add 1 to both sides.

$$-x^{2}+3x=-5.5+1$$

Add -5.5 and 1 to get -4.5.

$$-x^{2}+3x=-4.5$$

Divide both sides by -1.

$$\frac{-x^{2}+3x}{-1}=-\frac{4.5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{3}{-1}x=-\frac{4.5}{-1}$$

Divide 3 by -1.

$$x^{2}-3x=-\frac{4.5}{-1}$$

Divide -4.5 by -1.

$$x^{2}-3x=4.5$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=4.5+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=4.5+\frac{9}{4}$$

Add 4.5 to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=\frac{27}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{27}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{27}{4}}$$

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

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