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Solve 3x² + 4x - 11 = 0

The solutions of 3x² + 4x - 11 = 0 are x = (-2 + √37)/3 or x = (-2 - √37)/3, that is x ≈ 1.3609208 and x ≈ -2.6942542.

x² +x += 0
3x2+4x−11=0⇒x=−2+373, −2−3733x^2 + 4x - 11 = 0 \Rightarrow x = \frac{-2 + \sqrt{37}}{3},\ \frac{-2 - \sqrt{37}}{3}
Discriminant D = 148 · x ≈ 1.36092, -2.69425

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

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Graph of y = 3x² + 4x - 11-5-4-3-2-1123-20-101020304050x = 1.3609208x = -2.6942542vertexy = 3x² + 4x - 11
Graph of y = 3x² + 4x - 11. The curve crosses the x-axis at x = 1.3609208 and x = -2.6942542; the vertex is at (-0.66666667, -12.333333).

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 = 4, c = -11.

    3x2+4x−11=03x^2 + 4x - 11 = 0
  2. 2

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

    D=42−4⋅3⋅(−11)=16+132=148D = 4^2 - 4\cdot3\cdot(-11) = 16 + 132 = 148
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−4±1486x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{148}}{6}
  5. 5

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

    x=−2+373orx=−2−373x = \frac{-2 + \sqrt{37}}{3}\quad\text{or}\quad x = \frac{-2 - \sqrt{37}}{3}
  6. 6

    As decimals: x ≈ 1.3609208 and x ≈ -2.6942542.

  7. 7

    Check numerically: for x ≈ 1.3609208 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 4x - 11 = 0\ \Rightarrow\ x = \frac{-2 + \sqrt{37}}{3},\ \frac{-2 - \sqrt{37}}{3} \]
\[ 3x^2 + 4x - 11 = 0 \]
\[ D = 4^2 - 4\cdot3\cdot(-11) = 16 + 132 = 148 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{148}}{6} \]
\[ x = \frac{-2 + \sqrt{37}}{3}\quad\text{or}\quad x = \frac{-2 - \sqrt{37}}{3} \]

The parabola y = 3x² + 4x - 11

xy
-3.666714.667267
-2.6667-0.33293333
-1.6667-9.3331333
-0.6667-12.333333
0.3333-9.3335333
1.3333-0.33373333
2.333314.666067

Questions

What are the solutions of 3x² + 4x - 11 = 0?

The solutions of 3x² + 4x - 11 = 0 are x = (-2 + √37)/3 or x = (-2 - √37)/3, that is x ≈ 1.3609208 and x ≈ -2.6942542.

Does 3x² + 4x - 11 = 0 factor?

Not over the integers, because the discriminant 148 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 4x - 11?

(-2/3, -37/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -4/3, product = c/a = -11/3.

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    Solve 3x² + 4x - 11 = 0: x = (-2 + √37)/3 or x = (-2 - √37)/3 (steps) | WordGPT