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Solve -3x² - 11x - 7 = 0

The solutions of -3x² - 11x - 7 = 0 are x = (-11 + √37)/6 or x = (-11 - √37)/6, that is x ≈ -2.8471271 and x ≈ -0.81953958.

x² +x += 0
−3x2−11x−7=0⇒x=−11+376, −11−376-3x^2 - 11x - 7 = 0 \Rightarrow x = \frac{-11 + \sqrt{37}}{6},\ \frac{-11 - \sqrt{37}}{6}
Discriminant D = 37 · x ≈ -2.84713, -0.81954

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

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Graph of y = -3x² - 11x - 7-5-4-3-2-11-35-30-25-20-15-10-55x = -2.8471271x = -0.81953958vertexy = -3x² - 11x - 7
Graph of y = -3x² - 11x - 7. The curve crosses the x-axis at x = -2.8471271 and x = -0.81953958; the vertex is at (-1.8333333, 3.0833333).

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 = -7.

    −3x2−11x−7=0-3x^2 - 11x - 7 = 0
  2. 2

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

    D=(−11)2−4⋅(−3)⋅(−7)=121−84=37D = (-11)^2 - 4\cdot(-3)\cdot(-7) = 121 - 84 = 37
  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±37−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{37}}{-6}
  5. 5

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

    x=−11+376orx=−11−376x = \frac{-11 + \sqrt{37}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{37}}{6}
  6. 6

    As decimals: x ≈ -2.8471271 and x ≈ -0.81953958.

  7. 7

    Check numerically: for x ≈ -2.8471271 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 11x - 7 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{37}}{6},\ \frac{-11 - \sqrt{37}}{6} \]
\[ -3x^2 - 11x - 7 = 0 \]
\[ D = (-11)^2 - 4\cdot(-3)\cdot(-7) = 121 - 84 = 37 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{37}}{-6} \]
\[ x = \frac{-11 + \sqrt{37}}{6}\quad\text{or}\quad x = \frac{-11 - \sqrt{37}}{6} \]

The parabola y = -3x² - 11x - 7

xy
-4.8333-23.916067
-3.8333-8.9162667
-2.83330.08353333
-1.83333.0833333
-0.83330.08313333
0.1667-8.9170667
1.1667-23.917267

Questions

What are the solutions of -3x² - 11x - 7 = 0?

The solutions of -3x² - 11x - 7 = 0 are x = (-11 + √37)/6 or x = (-11 - √37)/6, that is x ≈ -2.8471271 and x ≈ -0.81953958.

Does -3x² - 11x - 7 = 0 factor?

Not over the integers, because the discriminant 37 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² - 11x - 7?

(-11/6, 37/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -11/3, product = c/a = 7/3.

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    Solve -3x² - 11x - 7 = 0: x = (-11 + √37)/6 or x = (-11 - √37)/6 (steps) | WordGPT