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Solve 3x² + 8x + 5 = 0

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

x² +x += 0
3x2+8x+5=0⇒x=−1, −533x^2 + 8x + 5 = 0 \Rightarrow x = -1,\ -\frac{5}{3}
Discriminant D = 4 · x ≈ -1, -1.66667

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

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Graph of y = 3x² + 8x + 5-4-3-2-11510152025x = -1x = -1.6666667vertexy = 3x² + 8x + 5
Graph of y = 3x² + 8x + 5. The curve crosses the x-axis at x = -1 and x = -1.6666667; the vertex is at (-1.3333333, -0.33333333).

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 = 5.

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

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

    D=82−4⋅3⋅5=64−60=4D = 8^2 - 4\cdot3\cdot5 = 64 - 60 = 4
  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+5)=0\left(x + 1\right)\left(3x + 5\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+5=0⇒x=−53x + 1 = 0 \Rightarrow x = -1\qquad 3x + 5 = 0 \Rightarrow x = -\frac{5}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±46x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{4}}{6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−53x = -1\quad\text{or}\quad x = -\frac{5}{3}
  8. 8

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

LaTeX source of the formulas on this page
\[ 3x^2 + 8x + 5 = 0\ \Rightarrow\ x = -1,\ -\frac{5}{3} \]
\[ 3x^2 + 8x + 5 = 0 \]
\[ D = 8^2 - 4\cdot3\cdot5 = 64 - 60 = 4 \]
\[ \left(x + 1\right)\left(3x + 5\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x + 5 = 0 \Rightarrow x = -\frac{5}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{4}}{6} \]
\[ x = -1\quad\text{or}\quad x = -\frac{5}{3} \]

The parabola y = 3x² + 8x + 5

xy
-4.333326.666067
-3.333311.666267
-2.33332.6664667
-1.3333-0.33333333
-0.33332.6668667
0.666711.667067
1.666726.667267

Questions

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

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

Does 3x² + 8x + 5 = 0 factor?

Yes: (x+1)(3x+5) = 0.

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

(-4/3, -1/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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