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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 ≈ 0.38196601 and x ≈ 2.618034.

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 ≈ 0.381966, 2.61803

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

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Graph of y = -x² + 3x - 1-2-112345-12-10-8-6-4-22x = 0.38196601x = 2.618034vertexy = -x² + 3x - 1
Graph of y = -x² + 3x - 1. The curve crosses the x-axis at x = 0.38196601 and x = 2.618034; 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=32−4⋅(−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 ≈ 0.38196601 and x ≈ 2.618034.

  7. 7

    Check numerically: for x ≈ 0.38196601 the left side is -1.110223e-16 (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
-1.5-7.75
-0.5-2.75
0.50.25
1.51.25
2.50.25
3.5-2.75
4.5-7.75

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 ≈ 0.38196601 and x ≈ 2.618034.

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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