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Solve -x² + 5x - 3 = 0

The solutions of -x² + 5x - 3 = 0 are x = (5 + √13)/2 or x = (5 - √13)/2, that is x ≈ 0.69722436 and x ≈ 4.3027756.

x² +x += 0
−x2+5x−3=0⇒x=5+132, 5−132-x^2 + 5x - 3 = 0 \Rightarrow x = \frac{5 + \sqrt{13}}{2},\ \frac{5 - \sqrt{13}}{2}
Discriminant D = 13 · x ≈ 0.697224, 4.30278

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

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Graph of y = -x² + 5x - 3-1123456-15-10-55x = 0.69722436x = 4.3027756vertexy = -x² + 5x - 3
Graph of y = -x² + 5x - 3. The curve crosses the x-axis at x = 0.69722436 and x = 4.3027756; the vertex is at (2.5, 3.25).

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 = 5, c = -3.

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

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

    D=52−4⋅(−1)⋅(−3)=25−12=13D = 5^2 - 4\cdot(-1)\cdot(-3) = 25 - 12 = 13
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−5±13−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{13}}{-2}
  5. 5

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

    x=5+132orx=5−132x = \frac{5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{13}}{2}
  6. 6

    As decimals: x ≈ 0.69722436 and x ≈ 4.3027756.

  7. 7

    Check numerically: for x ≈ 0.69722436 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 5x - 3 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{13}}{2},\ \frac{5 - \sqrt{13}}{2} \]
\[ -x^2 + 5x - 3 = 0 \]
\[ D = 5^2 - 4\cdot(-1)\cdot(-3) = 25 - 12 = 13 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{13}}{-2} \]
\[ x = \frac{5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{13}}{2} \]

The parabola y = -x² + 5x - 3

xy
-0.5-5.75
0.5-0.75
1.52.25
2.53.25
3.52.25
4.5-0.75
5.5-5.75

Questions

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

The solutions of -x² + 5x - 3 = 0 are x = (5 + √13)/2 or x = (5 - √13)/2, that is x ≈ 0.69722436 and x ≈ 4.3027756.

Does -x² + 5x - 3 = 0 factor?

Not over the integers, because the discriminant 13 is not a perfect square. Use the quadratic formula.

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

(5/2, 13/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = 3.

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