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Solve -2x² + 5x - 12 = 0

-2x² + 5x - 12 = 0 has no real solutions. The two complex solutions are x = (5 + √71 i)/4 and x = (5 - √71 i)/4.

x² +x += 0
−2x2+5x−12=0⇒x=5+71 i4, 5−71 i4 (complex)-2x^2 + 5x - 12 = 0 \Rightarrow x = \frac{5 + \sqrt{71}\, i}{4},\ \frac{5 - \sqrt{71}\, i}{4}\ \text{(complex)}
Discriminant D = -71

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

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Graph of y = -2x² + 5x - 12-2-112345-40-35-30-25-20-15-10-5vertexy = -2x² + 5x - 12
Graph of y = -2x² + 5x - 12. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (1.25, -8.875).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 5, c = -12.

    −2x2+5x−12=0-2x^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⋅(−2)⋅(−12)=25−96=−71D = 5^2 - 4\cdot(-2)\cdot(-12) = 25 - 96 = -71
  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±−71−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-71}}{-4}
  5. 5

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

    x=5+71 i4orx=5−71 i4x = \frac{5 + \sqrt{71}\, i}{4}\quad\text{or}\quad x = \frac{5 - \sqrt{71}\, i}{4}
LaTeX source of the formulas on this page
\[ -2x^2 + 5x - 12 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{71}\, i}{4},\ \frac{5 - \sqrt{71}\, i}{4} \]
\[ -2x^2 + 5x - 12 = 0 \]
\[ D = 5^2 - 4\cdot(-2)\cdot(-12) = 25 - 96 = -71 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-71}}{-4} \]
\[ x = \frac{5 + \sqrt{71}\, i}{4}\quad\text{or}\quad x = \frac{5 - \sqrt{71}\, i}{4} \]

The parabola y = -2x² + 5x - 12

xy
-1.75-26.875
-0.75-16.875
0.25-10.875
1.25-8.875
2.25-10.875
3.25-16.875
4.25-26.875

Questions

What are the solutions of -2x² + 5x - 12 = 0?

-2x² + 5x - 12 = 0 has no real solutions. The two complex solutions are x = (5 + √71 i)/4 and x = (5 - √71 i)/4.

Does -2x² + 5x - 12 = 0 factor?

Not over the integers, because the discriminant -71 is negative. Use the quadratic formula.

What is the vertex of y = -2x² + 5x - 12?

(5/4, -71/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5/2, product = c/a = 6.

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