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Solve 3x² + 6x - 9 = 0

The solutions of 3x² + 6x - 9 = 0 are x = 1 or x = -3.

x² +x += 0
3x2+6x−9=0⇒x=1, −33x^2 + 6x - 9 = 0 \Rightarrow x = 1,\ -3
Discriminant D = 144 · x ≈ 1, -3

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

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Graph of y = 3x² + 6x - 9-5-4-3-2-1123-20-101020304050x = 1x = -3vertexy = 3x² + 6x - 9
Graph of y = 3x² + 6x - 9. The curve crosses the x-axis at x = 1 and x = -3; the vertex is at (-1, -12).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = 6, c = -9.

    3x2+6x−9=03x^2 + 6x - 9 = 0
  2. 2

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

    D=62−4⋅3⋅(−9)=36+108=144D = 6^2 - 4\cdot3\cdot(-9) = 36 + 108 = 144
  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:

    3(x−1)(x+3)=03\left(x - 1\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−1=0⇒x=1x+3=0⇒x=−3x - 1 = 0 \Rightarrow x = 1\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−6±1446x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{144}}{6}
  7. 7

    Simplify to get the solutions:

    x=1orx=−3x = 1\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 3·(1) + 6·(1) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 6x - 9 = 0\ \Rightarrow\ x = 1,\ -3 \]
\[ 3x^2 + 6x - 9 = 0 \]
\[ D = 6^2 - 4\cdot3\cdot(-9) = 36 + 108 = 144 \]
\[ 3\left(x - 1\right)\left(x + 3\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{144}}{6} \]
\[ x = 1\quad\text{or}\quad x = -3 \]

The parabola y = 3x² + 6x - 9

xy
-415
-30
-2-9
-1-12
0-9
10
215

Questions

What are the solutions of 3x² + 6x - 9 = 0?

The solutions of 3x² + 6x - 9 = 0 are x = 1 or x = -3.

Does 3x² + 6x - 9 = 0 factor?

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

What is the vertex of y = 3x² + 6x - 9?

(-1, -12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = -3.

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