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Solve -x² + 5x - 12 = 0

-x² + 5x - 12 = 0 has no real solutions. The two complex solutions are x = (5 + √23 i)/2 and x = (5 - √23 i)/2.

x² +x += 0
−x2+5x−12=0⇒x=5+23 i2, 5−23 i2 (complex)-x^2 + 5x - 12 = 0 \Rightarrow x = \frac{5 + \sqrt{23}\, i}{2},\ \frac{5 - \sqrt{23}\, i}{2}\ \text{(complex)}
Discriminant D = -23

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

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Graph of y = -x² + 5x - 12-1123456-22-20-18-16-14-12-10-8-6-4vertexy = -x² + 5x - 12
Graph of y = -x² + 5x - 12. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (2.5, -5.75).

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 = 5, c = -12.

    −x2+5x−12=0-x^2 + 5x - 12 = 0
  2. 2

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

    D=52−4⋅(−1)⋅(−12)=25−48=−23D = 5^2 - 4\cdot(-1)\cdot(-12) = 25 - 48 = -23
  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=−5±−23−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-23}}{-2}
  5. 5

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

    x=5+23 i2orx=5−23 i2x = \frac{5 + \sqrt{23}\, i}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{23}\, i}{2}
LaTeX source of the formulas on this page
\[ -x^2 + 5x - 12 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{23}\, i}{2},\ \frac{5 - \sqrt{23}\, i}{2} \]
\[ -x^2 + 5x - 12 = 0 \]
\[ D = 5^2 - 4\cdot(-1)\cdot(-12) = 25 - 48 = -23 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-23}}{-2} \]
\[ x = \frac{5 + \sqrt{23}\, i}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{23}\, i}{2} \]

The parabola y = -x² + 5x - 12

xy
-0.5-14.75
0.5-9.75
1.5-6.75
2.5-5.75
3.5-6.75
4.5-9.75
5.5-14.75

Questions

What are the solutions of -x² + 5x - 12 = 0?

-x² + 5x - 12 = 0 has no real solutions. The two complex solutions are x = (5 + √23 i)/2 and x = (5 - √23 i)/2.

Does -x² + 5x - 12 = 0 factor?

Not over the integers, because the discriminant -23 is negative. Use the quadratic formula.

What is the vertex of y = -x² + 5x - 12?

(5/2, -23/4). 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 = 12.

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