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Solve -3x² - 6x + 11 = 0

The solutions of -3x² - 6x + 11 = 0 are x = (-3 + √42)/3 or x = (-3 - √42)/3, that is x ≈ -3.1602469 and x ≈ 1.1602469.

x² +x += 0
−3x2−6x+11=0⇒x=−3+423, −3−423-3x^2 - 6x + 11 = 0 \Rightarrow x = \frac{-3 + \sqrt{42}}{3},\ \frac{-3 - \sqrt{42}}{3}
Discriminant D = 168 · x ≈ -3.16025, 1.16025

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

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Graph of y = -3x² - 6x + 11-5-4-3-2-1123-50-40-30-20-101020x = -3.1602469x = 1.1602469vertexy = -3x² - 6x + 11
Graph of y = -3x² - 6x + 11. The curve crosses the x-axis at x = -3.1602469 and x = 1.1602469; the vertex is at (-1, 14).

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 = 11.

    −3x2−6x+11=0-3x^2 - 6x + 11 = 0
  2. 2

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

    D=(−6)2−4⋅(−3)⋅11=36+132=168D = (-6)^2 - 4\cdot(-3)\cdot11 = 36 + 132 = 168
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=6±168−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{168}}{-6}
  5. 5

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

    x=−3+423orx=−3−423x = \frac{-3 + \sqrt{42}}{3}\quad\text{or}\quad x = \frac{-3 - \sqrt{42}}{3}
  6. 6

    As decimals: x ≈ -3.1602469 and x ≈ 1.1602469.

  7. 7

    Check numerically: for x ≈ -3.1602469 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 6x + 11 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{42}}{3},\ \frac{-3 - \sqrt{42}}{3} \]
\[ -3x^2 - 6x + 11 = 0 \]
\[ D = (-6)^2 - 4\cdot(-3)\cdot11 = 36 + 132 = 168 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{168}}{-6} \]
\[ x = \frac{-3 + \sqrt{42}}{3}\quad\text{or}\quad x = \frac{-3 - \sqrt{42}}{3} \]

The parabola y = -3x² - 6x + 11

xy
-4-13
-32
-211
-114
011
12
2-13

Questions

What are the solutions of -3x² - 6x + 11 = 0?

The solutions of -3x² - 6x + 11 = 0 are x = (-3 + √42)/3 or x = (-3 - √42)/3, that is x ≈ -3.1602469 and x ≈ 1.1602469.

Does -3x² - 6x + 11 = 0 factor?

Not over the integers, because the discriminant 168 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² - 6x + 11?

(-1, 14). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = -11/3.

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    Solve -3x² - 6x + 11 = 0: x = (-3 + √42)/3 or x = (-3 - √42)/3 (steps) | WordGPT