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Solve 3x² + 11x - 2 = 0

The solutions of 3x² + 11x - 2 = 0 are x = (-11 + √145)/6 or x = (-11 - √145)/6, that is x ≈ 0.1735991 and x ≈ -3.8402658.

x² +x += 0
3x2+11x−2=0⇒x=−11+1456, −11−14563x^2 + 11x - 2 = 0 \Rightarrow x = \frac{-11 + \sqrt{145}}{6},\ \frac{-11 - \sqrt{145}}{6}
Discriminant D = 145 · x ≈ 0.173599, -3.84027

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

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Graph of y = 3x² + 11x - 2-6-5-4-3-2-112-20-101020304050x = 0.1735991x = -3.8402658vertexy = 3x² + 11x - 2
Graph of y = 3x² + 11x - 2. The curve crosses the x-axis at x = 0.1735991 and x = -3.8402658; the vertex is at (-1.8333333, -12.083333).

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 = 11, c = -2.

    3x2+11x−2=03x^2 + 11x - 2 = 0
  2. 2

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

    D=112−4⋅3⋅(−2)=121+24=145D = 11^2 - 4\cdot3\cdot(-2) = 121 + 24 = 145
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1456x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{145}}{6}
  5. 5

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

    x=−11+1456orx=−11−1456x = \frac{-11 + \sqrt{145}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{145}}{6}
  6. 6

    As decimals: x ≈ 0.1735991 and x ≈ -3.8402658.

  7. 7

    Check numerically: for x ≈ 0.1735991 the left side is 4.4408921e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 11x - 2 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{145}}{6},\ \frac{-11 - \sqrt{145}}{6} \]
\[ 3x^2 + 11x - 2 = 0 \]
\[ D = 11^2 - 4\cdot3\cdot(-2) = 121 + 24 = 145 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{145}}{6} \]
\[ x = \frac{-11 + \sqrt{145}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{145}}{6} \]

The parabola y = 3x² + 11x - 2

xy
-4.833314.916067
-3.8333-0.08373333
-2.8333-9.0835333
-1.8333-12.083333
-0.8333-9.0831333
0.1667-0.08293333
1.166714.917267

Questions

What are the solutions of 3x² + 11x - 2 = 0?

The solutions of 3x² + 11x - 2 = 0 are x = (-11 + √145)/6 or x = (-11 - √145)/6, that is x ≈ 0.1735991 and x ≈ -3.8402658.

Does 3x² + 11x - 2 = 0 factor?

Not over the integers, because the discriminant 145 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 11x - 2?

(-11/6, -145/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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    Solve 3x² + 11x - 2 = 0: x = (-11 + √145)/6 or x = (-11 - √145)/6 (steps) | WordGPT