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Solve -3x² + 9x - 6 = 0

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

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

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

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Graph of y = -3x² + 9x - 6-11234-25-20-15-10-5x = 1x = 2vertexy = -3x² + 9x - 6
Graph of y = -3x² + 9x - 6. The curve crosses the x-axis at x = 1 and x = 2; 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=0-3x^2 + 9x - 6 = 0
  2. 2

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

    D=92−4⋅(−3)⋅(−6)=81−72=9D = 9^2 - 4\cdot(-3)\cdot(-6) = 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−1)(x−2)=0-3\left(x - 1\right)\left(x - 2\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−2=0⇒x=2x - 1 = 0 \Rightarrow x = 1\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

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

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

The parabola y = -3x² + 9x - 6

xy
-1.5-26.25
-0.5-11.25
0.5-2.25
1.50.75
2.5-2.25
3.5-11.25
4.5-26.25

Questions

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

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

Does -3x² + 9x - 6 = 0 factor?

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

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

(3/2, 3/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = 2.

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