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Solve -x² + 3x - 5 = 0

-x² + 3x - 5 = 0 has no real solutions. The two complex solutions are x = (3 + √11 i)/2 and x = (3 - √11 i)/2.

x² +x += 0
−x2+3x−5=0⇒x=3+11 i2, 3−11 i2 (complex)-x^2 + 3x - 5 = 0 \Rightarrow x = \frac{3 + \sqrt{11}\, i}{2},\ \frac{3 - \sqrt{11}\, i}{2}\ \text{(complex)}
Discriminant D = -11

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

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Graph of y = -x² + 3x - 5-2-112345-18-16-14-12-10-8-6-4-2vertexy = -x² + 3x - 5
Graph of y = -x² + 3x - 5. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (1.5, -2.75).

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 = -5.

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

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

    D=32−4⋅(−1)⋅(−5)=9−20=−11D = 3^2 - 4\cdot(-1)\cdot(-5) = 9 - 20 = -11
  3. 3

    D is negative, so there are no real solutions; the two solutions are complex numbers.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−3±−11−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{-11}}{-2}
  5. 5

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

    x=3+11 i2orx=3−11 i2x = \frac{3 + \sqrt{11}\, i}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{11}\, i}{2}
LaTeX source of the formulas on this page
\[ -x^2 + 3x - 5 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{11}\, i}{2},\ \frac{3 - \sqrt{11}\, i}{2} \]
\[ -x^2 + 3x - 5 = 0 \]
\[ D = 3^2 - 4\cdot(-1)\cdot(-5) = 9 - 20 = -11 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{-11}}{-2} \]
\[ x = \frac{3 + \sqrt{11}\, i}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{11}\, i}{2} \]

The parabola y = -x² + 3x - 5

xy
-1.5-11.75
-0.5-6.75
0.5-3.75
1.5-2.75
2.5-3.75
3.5-6.75
4.5-11.75

Questions

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

-x² + 3x - 5 = 0 has no real solutions. The two complex solutions are x = (3 + √11 i)/2 and x = (3 - √11 i)/2.

Does -x² + 3x - 5 = 0 factor?

Not over the integers, because the discriminant -11 is negative. Use the quadratic formula.

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

(3/2, -11/4). 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 = 5.

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