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Solve 3x² + 5x - 11 = 0

The solutions of 3x² + 5x - 11 = 0 are x = (-5 + √157)/6 or x = (-5 - √157)/6, that is x ≈ 1.254994 and x ≈ -2.9216607.

x² +x += 0
3x2+5x−11=0⇒x=−5+1576, −5−15763x^2 + 5x - 11 = 0 \Rightarrow x = \frac{-5 + \sqrt{157}}{6},\ \frac{-5 - \sqrt{157}}{6}
Discriminant D = 157 · x ≈ 1.25499, -2.92166

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

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Graph of y = 3x² + 5x - 11-5-4-3-2-1123-20-101020304050x = 1.254994x = -2.9216607vertexy = 3x² + 5x - 11
Graph of y = 3x² + 5x - 11. The curve crosses the x-axis at x = 1.254994 and x = -2.9216607; the vertex is at (-0.83333333, -13.083333).

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 = 5, c = -11.

    3x2+5x−11=03x^2 + 5x - 11 = 0
  2. 2

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

    D=52−4⋅3⋅(−11)=25+132=157D = 5^2 - 4\cdot3\cdot(-11) = 25 + 132 = 157
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−5±1576x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{157}}{6}
  5. 5

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

    x=−5+1576orx=−5−1576x = \frac{-5 + \sqrt{157}}{6}\quad\text{or}\quad x = \frac{-5 - \sqrt{157}}{6}
  6. 6

    As decimals: x ≈ 1.254994 and x ≈ -2.9216607.

  7. 7

    Check numerically: for x ≈ 1.254994 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 5x - 11 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{157}}{6},\ \frac{-5 - \sqrt{157}}{6} \]
\[ 3x^2 + 5x - 11 = 0 \]
\[ D = 5^2 - 4\cdot3\cdot(-11) = 25 + 132 = 157 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{157}}{6} \]
\[ x = \frac{-5 + \sqrt{157}}{6}\quad\text{or}\quad x = \frac{-5 - \sqrt{157}}{6} \]

The parabola y = 3x² + 5x - 11

xy
-3.833313.916067
-2.8333-1.0837333
-1.8333-10.083533
-0.8333-13.083333
0.1667-10.083133
1.1667-1.0829333
2.166713.917267

Questions

What are the solutions of 3x² + 5x - 11 = 0?

The solutions of 3x² + 5x - 11 = 0 are x = (-5 + √157)/6 or x = (-5 - √157)/6, that is x ≈ 1.254994 and x ≈ -2.9216607.

Does 3x² + 5x - 11 = 0 factor?

Not over the integers, because the discriminant 157 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 5x - 11?

(-5/6, -157/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5/3, product = c/a = -11/3.

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    Solve 3x² + 5x - 11 = 0: x = (-5 + √157)/6 or x = (-5 - √157)/6 (steps) | WordGPT