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Solve 3x² + 11x - 5 = 0

The solutions of 3x² + 11x - 5 = 0 are x = (-11 + √181)/6 or x = (-11 - √181)/6, that is x ≈ 0.40893734 and x ≈ -4.075604.

x² +x += 0
3x2+11x−5=0⇒x=−11+1816, −11−18163x^2 + 11x - 5 = 0 \Rightarrow x = \frac{-11 + \sqrt{181}}{6},\ \frac{-11 - \sqrt{181}}{6}
Discriminant D = 181 · x ≈ 0.408937, -4.0756

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

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Graph of y = 3x² + 11x - 5-6-5-4-3-2-112-20-101020304050x = 0.40893734x = -4.075604vertexy = 3x² + 11x - 5
Graph of y = 3x² + 11x - 5. The curve crosses the x-axis at x = 0.40893734 and x = -4.075604; the vertex is at (-1.8333333, -15.083333).

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 = -5.

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

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

    D=112−4⋅3⋅(−5)=121+60=181D = 11^2 - 4\cdot3\cdot(-5) = 121 + 60 = 181
  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±1816x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{181}}{6}
  5. 5

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

    x=−11+1816orx=−11−1816x = \frac{-11 + \sqrt{181}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{181}}{6}
  6. 6

    As decimals: x ≈ 0.40893734 and x ≈ -4.075604.

  7. 7

    Check numerically: for x ≈ 0.40893734 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 11x - 5 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{181}}{6},\ \frac{-11 - \sqrt{181}}{6} \]
\[ 3x^2 + 11x - 5 = 0 \]
\[ D = 11^2 - 4\cdot3\cdot(-5) = 121 + 60 = 181 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{181}}{6} \]
\[ x = \frac{-11 + \sqrt{181}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{181}}{6} \]

The parabola y = 3x² + 11x - 5

xy
-4.833311.916067
-3.8333-3.0837333
-2.8333-12.083533
-1.8333-15.083333
-0.8333-12.083133
0.1667-3.0829333
1.166711.917267

Questions

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

The solutions of 3x² + 11x - 5 = 0 are x = (-11 + √181)/6 or x = (-11 - √181)/6, that is x ≈ 0.40893734 and x ≈ -4.075604.

Does 3x² + 11x - 5 = 0 factor?

Not over the integers, because the discriminant 181 is not a perfect square. Use the quadratic formula.

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

(-11/6, -181/12). 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 = -5/3.

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    Solve 3x² + 11x - 5 = 0: x = (-11 + √181)/6 or x = (-11 - √181)/6 (steps) | WordGPT