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Solve -x² + 9 = 0

The solutions of -x² + 9 = 0 are x = -3 or x = 3.

x² +x += 0
−x2+9=0⇒x=−3, 3-x^2 + 9 = 0 \Rightarrow x = -3,\ 3
Discriminant D = 36 · x ≈ -3, 3

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

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Graph of y = -x² + 9-4-224-20-15-10-5510x = -3x = 3vertexy = -x² + 9
Graph of y = -x² + 9. The curve crosses the x-axis at x = -3 and x = 3; the vertex is at (0, 9).

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 = 9.

    −x2+9=0-x^2 + 9 = 0
  2. 2

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

    D=02−4⋅(−1)⋅9=0+36=36D = 0^2 - 4\cdot(-1)\cdot9 = 0 + 36 = 36
  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+3)(x−3)=0-\left(x + 3\right)\left(x - 3\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+3=0⇒x=−3x−3=0⇒x=3x + 3 = 0 \Rightarrow x = -3\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=0±36−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{36}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−3orx=3x = -3\quad\text{or}\quad x = 3
  8. 8

    Check by putting x = -3 back in: a·x² + b·x + c = -1·(9) + 0·(-3) + 9 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 9 = 0\ \Rightarrow\ x = -3,\ 3 \]
\[ -x^2 + 9 = 0 \]
\[ D = 0^2 - 4\cdot(-1)\cdot9 = 0 + 36 = 36 \]
\[ -\left(x + 3\right)\left(x - 3\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{36}}{-2} \]
\[ x = -3\quad\text{or}\quad x = 3 \]

The parabola y = -x² + 9

xy
-30
-25
-18
09
18
25
30

Questions

What are the solutions of -x² + 9 = 0?

The solutions of -x² + 9 = 0 are x = -3 or x = 3.

Does -x² + 9 = 0 factor?

Yes: -1(x+3)(x-3) = 0.

What is the vertex of y = -x² + 9?

(0, 9). 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 = -9.

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