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Solve -x² + 1 = 0

The solutions of -x² + 1 = 0 are x = -1 or x = 1.

x² +x += 0
−x2+1=0⇒x=−1, 1-x^2 + 1 = 0 \Rightarrow x = -1,\ 1
Discriminant D = 4 · x ≈ -1, 1

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

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Graph of y = -x² + 1-3-2-1123-12-10-8-6-4-22x = -1x = 1vertexy = -x² + 1
Graph of y = -x² + 1. The curve crosses the x-axis at x = -1 and x = 1; the vertex is at (0, 1).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 0, c = 1.

    −x2+1=0-x^2 + 1 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=02−4⋅(−1)⋅1=0+4=4D = 0^2 - 4\cdot(-1)\cdot1 = 0 + 4 = 4
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x+1)(x−1)=0-\left(x + 1\right)\left(x - 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+1=0⇒x=−1x−1=0⇒x=1x + 1 = 0 \Rightarrow x = -1\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=0±4−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{4}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−1orx=1x = -1\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -1·(1) + 0·(-1) + 1 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 1 = 0\ \Rightarrow\ x = -1,\ 1 \]
\[ -x^2 + 1 = 0 \]
\[ D = 0^2 - 4\cdot(-1)\cdot1 = 0 + 4 = 4 \]
\[ -\left(x + 1\right)\left(x - 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{4}}{-2} \]
\[ x = -1\quad\text{or}\quad x = 1 \]

The parabola y = -x² + 1

xy
-3-8
-2-3
-10
01
10
2-3
3-8

Questions

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

The solutions of -x² + 1 = 0 are x = -1 or x = 1.

Does -x² + 1 = 0 factor?

Yes: -1(x+1)(x-1) = 0.

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.

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