Solve -x² - 1 = 0

-x² - 1 = 0 has no real solutions. The two complex solutions are x = i and x = -i.

x² +x += 0

-x^2 - 1 = 0 \Rightarrow x = i,\ -i\ \text{(complex)}

Discriminant D = -4

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

-x^2 - 1 = 0\ \Rightarrow\ x = i,\ -i

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Graph of y = -x² - 1. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0, -1).

How to work it out

1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 0, c = -1.
$-x^2 - 1 = 0$
2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
$D = 0^2 - 4\cdot(-1)\cdot(-1) = 0 - 4 = -4$
3
D is negative, so there are no real solutions; the two solutions are complex numbers.
4
The quadratic formula works for every quadratic:
$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{-4}}{-2}$
5
Simplify (pull the square out of the root) to get the solutions:
$x = i\quad\text{or}\quad x = -i$

LaTeX source of the formulas on this page

\[ -x^2 - 1 = 0\ \Rightarrow\ x = i,\ -i \]
\[ -x^2 - 1 = 0 \]
\[ D = 0^2 - 4\cdot(-1)\cdot(-1) = 0 - 4 = -4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{-4}}{-2} \]
\[ x = i\quad\text{or}\quad x = -i \]

The parabola y = -x² - 1

x	y
-3	-10
-2	-5
-1	-2
0	-1
1	-2
2	-5
3	-10

Questions

What are the solutions of -x² - 1 = 0?

-x² - 1 = 0 has no real solutions. The two complex solutions are x = i and x = -i.

Does -x² - 1 = 0 factor?

Not over the integers, because the discriminant -4 is negative. Use the quadratic formula.

What is the vertex of y = -x² - 1?

(0, -1). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = 1.

Solve -x² - 1 = 0

How to work it out

The parabola y = -x² - 1

Questions

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