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Solve -3x² - 10x - 6 = 0

The solutions of -3x² - 10x - 6 = 0 are x = (-5 + √7)/3 or x = (-5 - √7)/3, that is x ≈ -2.5485838 and x ≈ -0.78474956.

x² +x += 0
−3x2−10x−6=0⇒x=−5+73, −5−73-3x^2 - 10x - 6 = 0 \Rightarrow x = \frac{-5 + \sqrt{7}}{3},\ \frac{-5 - \sqrt{7}}{3}
Discriminant D = 28 · x ≈ -2.54858, -0.78475

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

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Graph of y = -3x² - 10x - 6-5-4-3-2-11-30-25-20-15-10-55x = -2.5485838x = -0.78474956vertexy = -3x² - 10x - 6
Graph of y = -3x² - 10x - 6. The curve crosses the x-axis at x = -2.5485838 and x = -0.78474956; the vertex is at (-1.6666667, 2.3333333).

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 = -10, c = -6.

    −3x2−10x−6=0-3x^2 - 10x - 6 = 0
  2. 2

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

    D=(−10)2−4⋅(−3)⋅(−6)=100−72=28D = (-10)^2 - 4\cdot(-3)\cdot(-6) = 100 - 72 = 28
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=10±28−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{28}}{-6}
  5. 5

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

    x=−5+73orx=−5−73x = \frac{-5 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{-5 - \sqrt{7}}{3}
  6. 6

    As decimals: x ≈ -2.5485838 and x ≈ -0.78474956.

  7. 7

    Check numerically: for x ≈ -2.5485838 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 10x - 6 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{7}}{3},\ \frac{-5 - \sqrt{7}}{3} \]
\[ -3x^2 - 10x - 6 = 0 \]
\[ D = (-10)^2 - 4\cdot(-3)\cdot(-6) = 100 - 72 = 28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{28}}{-6} \]
\[ x = \frac{-5 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{-5 - \sqrt{7}}{3} \]

The parabola y = -3x² - 10x - 6

xy
-4.6667-24.667267
-3.6667-9.6670667
-2.6667-0.66686667
-1.66672.3333333
-0.6667-0.66646667
0.3333-9.6662667
1.3333-24.666067

Questions

What are the solutions of -3x² - 10x - 6 = 0?

The solutions of -3x² - 10x - 6 = 0 are x = (-5 + √7)/3 or x = (-5 - √7)/3, that is x ≈ -2.5485838 and x ≈ -0.78474956.

Does -3x² - 10x - 6 = 0 factor?

Not over the integers, because the discriminant 28 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² - 10x - 6?

(-5/3, 7/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -10/3, product = c/a = 2.

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