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Solve -3x² + 3x + 2 = 0

The solutions of -3x² + 3x + 2 = 0 are x = (3 + √33)/6 or x = (3 - √33)/6, that is x ≈ -0.45742711 and x ≈ 1.4574271.

x² +x += 0
−3x2+3x+2=0⇒x=3+336, 3−336-3x^2 + 3x + 2 = 0 \Rightarrow x = \frac{3 + \sqrt{33}}{6},\ \frac{3 - \sqrt{33}}{6}
Discriminant D = 33 · x ≈ -0.457427, 1.45743

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

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Graph of y = -3x² + 3x + 2-2-1123-35-30-25-20-15-10-55x = -0.45742711x = 1.4574271vertexy = -3x² + 3x + 2
Graph of y = -3x² + 3x + 2. The curve crosses the x-axis at x = -0.45742711 and x = 1.4574271; the vertex is at (0.5, 2.75).

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 = 3, c = 2.

    −3x2+3x+2=0-3x^2 + 3x + 2 = 0
  2. 2

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

    D=32−4⋅(−3)⋅2=9+24=33D = 3^2 - 4\cdot(-3)\cdot2 = 9 + 24 = 33
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−3±33−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{33}}{-6}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=3+336orx=3−336x = \frac{3 + \sqrt{33}}{6}\quad\text{or}\quad x = \frac{3 - \sqrt{33}}{6}
  6. 6

    As decimals: x ≈ -0.45742711 and x ≈ 1.4574271.

  7. 7

    Check numerically: for x ≈ -0.45742711 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 3x + 2 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{33}}{6},\ \frac{3 - \sqrt{33}}{6} \]
\[ -3x^2 + 3x + 2 = 0 \]
\[ D = 3^2 - 4\cdot(-3)\cdot2 = 9 + 24 = 33 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{33}}{-6} \]
\[ x = \frac{3 + \sqrt{33}}{6}\quad\text{or}\quad x = \frac{3 - \sqrt{33}}{6} \]

The parabola y = -3x² + 3x + 2

xy
-2.5-24.25
-1.5-9.25
-0.5-0.25
0.52.75
1.5-0.25
2.5-9.25
3.5-24.25

Questions

What are the solutions of -3x² + 3x + 2 = 0?

The solutions of -3x² + 3x + 2 = 0 are x = (3 + √33)/6 or x = (3 - √33)/6, that is x ≈ -0.45742711 and x ≈ 1.4574271.

Does -3x² + 3x + 2 = 0 factor?

Not over the integers, because the discriminant 33 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² + 3x + 2?

(1/2, 11/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -2/3.

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    Solve -3x² + 3x + 2 = 0: x = (3 + √33)/6 or x = (3 - √33)/6 (steps) | WordGPT