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Solve -x² - 3x - 1 = 0

The solutions of -x² - 3x - 1 = 0 are x = (-3 + √5)/2 or x = (-3 - √5)/2, that is x ≈ -2.618034 and x ≈ -0.38196601.

x² +x += 0
−x2−3x−1=0⇒x=−3+52, −3−52-x^2 - 3x - 1 = 0 \Rightarrow x = \frac{-3 + \sqrt{5}}{2},\ \frac{-3 - \sqrt{5}}{2}
Discriminant D = 5 · x ≈ -2.61803, -0.381966

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

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Graph of y = -x² - 3x - 1-5-4-3-2-112-12-10-8-6-4-22x = -2.618034x = -0.38196601vertexy = -x² - 3x - 1
Graph of y = -x² - 3x - 1. The curve crosses the x-axis at x = -2.618034 and x = -0.38196601; the vertex is at (-1.5, 1.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -3, c = -1.

    −x2−3x−1=0-x^2 - 3x - 1 = 0
  2. 2

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

    D=(−3)2−4⋅(−1)⋅(−1)=9−4=5D = (-3)^2 - 4\cdot(-1)\cdot(-1) = 9 - 4 = 5
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±5−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{5}}{-2}
  5. 5

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

    x=−3+52orx=−3−52x = \frac{-3 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{5}}{2}
  6. 6

    As decimals: x ≈ -2.618034 and x ≈ -0.38196601.

  7. 7

    Check numerically: for x ≈ -2.618034 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 3x - 1 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{5}}{2},\ \frac{-3 - \sqrt{5}}{2} \]
\[ -x^2 - 3x - 1 = 0 \]
\[ D = (-3)^2 - 4\cdot(-1)\cdot(-1) = 9 - 4 = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{5}}{-2} \]
\[ x = \frac{-3 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{5}}{2} \]

The parabola y = -x² - 3x - 1

xy
-4.5-7.75
-3.5-2.75
-2.50.25
-1.51.25
-0.50.25
0.5-2.75
1.5-7.75

Questions

What are the solutions of -x² - 3x - 1 = 0?

The solutions of -x² - 3x - 1 = 0 are x = (-3 + √5)/2 or x = (-3 - √5)/2, that is x ≈ -2.618034 and x ≈ -0.38196601.

Does -x² - 3x - 1 = 0 factor?

Not over the integers, because the discriminant 5 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 3x - 1?

(-3/2, 5/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 1.

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