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Solve 3x² + 5x - 8 = 0

The solutions of 3x² + 5x - 8 = 0 are x = 1 or x = -8/3.

x² +x += 0
3x2+5x−8=0⇒x=1, −833x^2 + 5x - 8 = 0 \Rightarrow x = 1,\ -\frac{8}{3}
Discriminant D = 121 · x ≈ 1, -2.66667

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

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Graph of y = 3x² + 5x - 8-5-4-3-2-1123-101020304050x = 1x = -2.6666667vertexy = 3x² + 5x - 8
Graph of y = 3x² + 5x - 8. The curve crosses the x-axis at x = 1 and x = -2.6666667; the vertex is at (-0.83333333, -10.083333).

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 = 5, c = -8.

    3x2+5x−8=03x^2 + 5x - 8 = 0
  2. 2

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

    D=52−4⋅3⋅(−8)=25+96=121D = 5^2 - 4\cdot3\cdot(-8) = 25 + 96 = 121
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−1)(3x+8)=0\left(x - 1\right)\left(3x + 8\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−1=0⇒x=13x+8=0⇒x=−83x - 1 = 0 \Rightarrow x = 1\qquad 3x + 8 = 0 \Rightarrow x = -\frac{8}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±1216x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{121}}{6}
  7. 7

    Simplify to get the solutions:

    x=1orx=−83x = 1\quad\text{or}\quad x = -\frac{8}{3}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 3·(1) + 5·(1) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 5x - 8 = 0\ \Rightarrow\ x = 1,\ -\frac{8}{3} \]
\[ 3x^2 + 5x - 8 = 0 \]
\[ D = 5^2 - 4\cdot3\cdot(-8) = 25 + 96 = 121 \]
\[ \left(x - 1\right)\left(3x + 8\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x + 8 = 0 \Rightarrow x = -\frac{8}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{121}}{6} \]
\[ x = 1\quad\text{or}\quad x = -\frac{8}{3} \]

The parabola y = 3x² + 5x - 8

xy
-3.833316.916067
-2.83331.9162667
-1.8333-7.0835333
-0.8333-10.083333
0.1667-7.0831333
1.16671.9170667
2.166716.917267

Questions

What are the solutions of 3x² + 5x - 8 = 0?

The solutions of 3x² + 5x - 8 = 0 are x = 1 or x = -8/3.

Does 3x² + 5x - 8 = 0 factor?

Yes: (x-1)(3x+8) = 0.

What is the vertex of y = 3x² + 5x - 8?

(-5/6, -121/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5/3, product = c/a = -8/3.

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