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Solve 3x² - 9x + 6 = 0

The solutions of 3x² - 9x + 6 = 0 are x = 2 or x = 1.

x² +x += 0
3x2−9x+6=0⇒x=2, 13x^2 - 9x + 6 = 0 \Rightarrow x = 2,\ 1
Discriminant D = 9 · x ≈ 2, 1

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

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Graph of y = 3x² - 9x + 6-11234510152025x = 2x = 1vertexy = 3x² - 9x + 6
Graph of y = 3x² - 9x + 6. The curve crosses the x-axis at x = 2 and x = 1; the vertex is at (1.5, -0.75).

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 = -9, c = 6.

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

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

    D=(−9)2−4⋅3⋅6=81−72=9D = (-9)^2 - 4\cdot3\cdot6 = 81 - 72 = 9
  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−2)(x−1)=03\left(x - 2\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−2=0⇒x=2x−1=0⇒x=1x - 2 = 0 \Rightarrow x = 2\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±96x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{9}}{6}
  7. 7

    Simplify to get the solutions:

    x=2orx=1x = 2\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 3·(4) + (-9)·(2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 9x + 6 = 0\ \Rightarrow\ x = 2,\ 1 \]
\[ 3x^2 - 9x + 6 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot6 = 81 - 72 = 9 \]
\[ 3\left(x - 2\right)\left(x - 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{9}}{6} \]
\[ x = 2\quad\text{or}\quad x = 1 \]

The parabola y = 3x² - 9x + 6

xy
-1.526.25
-0.511.25
0.52.25
1.5-0.75
2.52.25
3.511.25
4.526.25

Questions

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

The solutions of 3x² - 9x + 6 = 0 are x = 2 or x = 1.

Does 3x² - 9x + 6 = 0 factor?

Yes: 3(x-2)(x-1) = 0.

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

(3/2, -3/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = 2.

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