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Solve -3x² + 8x + 5 = 0

The solutions of -3x² + 8x + 5 = 0 are x = (4 + √31)/3 or x = (4 - √31)/3, that is x ≈ -0.52258812 and x ≈ 3.1892548.

x² +x += 0
−3x2+8x+5=0⇒x=4+313, 4−313-3x^2 + 8x + 5 = 0 \Rightarrow x = \frac{4 + \sqrt{31}}{3},\ \frac{4 - \sqrt{31}}{3}
Discriminant D = 124 · x ≈ -0.522588, 3.18925

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

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Graph of y = -3x² + 8x + 5-3-2-112345-50-40-30-20-1010x = -0.52258812x = 3.1892548vertexy = -3x² + 8x + 5
Graph of y = -3x² + 8x + 5. The curve crosses the x-axis at x = -0.52258812 and x = 3.1892548; the vertex is at (1.3333333, 10.333333).

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=0-3x^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=124D = 8^2 - 4\cdot(-3)\cdot5 = 64 + 60 = 124
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±124−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{124}}{-6}
  5. 5

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

    x=4+313orx=4−313x = \frac{4 + \sqrt{31}}{3}\quad\text{or}\quad x = \frac{4 - \sqrt{31}}{3}
  6. 6

    As decimals: x ≈ -0.52258812 and x ≈ 3.1892548.

  7. 7

    Check numerically: for x ≈ -0.52258812 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 8x + 5 = 0\ \Rightarrow\ x = \frac{4 + \sqrt{31}}{3},\ \frac{4 - \sqrt{31}}{3} \]
\[ -3x^2 + 8x + 5 = 0 \]
\[ D = 8^2 - 4\cdot(-3)\cdot5 = 64 + 60 = 124 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{124}}{-6} \]
\[ x = \frac{4 + \sqrt{31}}{3}\quad\text{or}\quad x = \frac{4 - \sqrt{31}}{3} \]

The parabola y = -3x² + 8x + 5

xy
-1.6667-16.667267
-0.6667-1.6670667
0.33337.3331333
1.333310.333333
2.33337.3335333
3.3333-1.6662667
4.3333-16.666067

Questions

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

The solutions of -3x² + 8x + 5 = 0 are x = (4 + √31)/3 or x = (4 - √31)/3, that is x ≈ -0.52258812 and x ≈ 3.1892548.

Does -3x² + 8x + 5 = 0 factor?

Not over the integers, because the discriminant 124 is not a perfect square. Use the quadratic formula.

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

(4/3, 31/3). 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 = -5/3.

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    Solve -3x² + 8x + 5 = 0: x = (4 + √31)/3 or x = (4 - √31)/3 (steps) | WordGPT