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Solve -3x² + 8x - 7 = 0

-3x² + 8x - 7 = 0 has no real solutions. The two complex solutions are x = (4 + √5 i)/3 and x = (4 - √5 i)/3.

x² +x += 0
−3x2+8x−7=0⇒x=4+5 i3, 4−5 i3 (complex)-3x^2 + 8x - 7 = 0 \Rightarrow x = \frac{4 + \sqrt{5}\, i}{3},\ \frac{4 - \sqrt{5}\, i}{3}\ \text{(complex)}
Discriminant D = -20

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

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Graph of y = -3x² + 8x - 7-2-112345-50-40-30-20-10vertexy = -3x² + 8x - 7
Graph of y = -3x² + 8x - 7. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (1.3333333, -1.6666667).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = 8, c = -7.

    −3x2+8x−7=0-3x^2 + 8x - 7 = 0
  2. 2

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

    D=82−4⋅(−3)⋅(−7)=64−84=−20D = 8^2 - 4\cdot(-3)\cdot(-7) = 64 - 84 = -20
  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=−8±−20−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{-20}}{-6}
  5. 5

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

    x=4+5 i3orx=4−5 i3x = \frac{4 + \sqrt{5}\, i}{3}\quad\text{or}\quad x = \frac{4 - \sqrt{5}\, i}{3}
LaTeX source of the formulas on this page
\[ -3x^2 + 8x - 7 = 0\ \Rightarrow\ x = \frac{4 + \sqrt{5}\, i}{3},\ \frac{4 - \sqrt{5}\, i}{3} \]
\[ -3x^2 + 8x - 7 = 0 \]
\[ D = 8^2 - 4\cdot(-3)\cdot(-7) = 64 - 84 = -20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{-20}}{-6} \]
\[ x = \frac{4 + \sqrt{5}\, i}{3}\quad\text{or}\quad x = \frac{4 - \sqrt{5}\, i}{3} \]

The parabola y = -3x² + 8x - 7

xy
-1.6667-28.667267
-0.6667-13.667067
0.3333-4.6668667
1.3333-1.6666667
2.3333-4.6664667
3.3333-13.666267
4.3333-28.666067

Questions

What are the solutions of -3x² + 8x - 7 = 0?

-3x² + 8x - 7 = 0 has no real solutions. The two complex solutions are x = (4 + √5 i)/3 and x = (4 - √5 i)/3.

Does -3x² + 8x - 7 = 0 factor?

Not over the integers, because the discriminant -20 is negative. Use the quadratic formula.

What is the vertex of y = -3x² + 8x - 7?

(4/3, -5/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8/3, product = c/a = 7/3.

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