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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, −3x^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 - 9-4-224-10-55101520x = 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=0x^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\cdot1\cdot(-9) = 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±362x = \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
\[ x^2 - 9 = 0\ \Rightarrow\ x = 3,\ -3 \]
\[ x^2 - 9 = 0 \]
\[ D = 0^2 - 4\cdot1\cdot(-9) = 0 + 36 = 36 \]
\[ \left(x - 3\right)\left(x + 3\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{36}}{2} \]
\[ x = 3\quad\text{or}\quad x = -3 \]

The parabola y = x² - 9

xy
-30
-2-5
-1-8
0-9
1-8
2-5
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: (x-3)(x+3) = 0.

What is the vertex of y = x² - 9?

(0, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -9.

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