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Solve -2x² + 6x + 3 = 0

The solutions of -2x² + 6x + 3 = 0 are x = (3 + √15)/2 or x = (3 - √15)/2, that is x ≈ -0.43649167 and x ≈ 3.4364917.

x² +x += 0
−2x2+6x+3=0⇒x=3+152, 3−152-2x^2 + 6x + 3 = 0 \Rightarrow x = \frac{3 + \sqrt{15}}{2},\ \frac{3 - \sqrt{15}}{2}
Discriminant D = 60 · x ≈ -0.436492, 3.43649

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

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Graph of y = -2x² + 6x + 3-2-112345-30-20-1010x = -0.43649167x = 3.4364917vertexy = -2x² + 6x + 3
Graph of y = -2x² + 6x + 3. The curve crosses the x-axis at x = -0.43649167 and x = 3.4364917; the vertex is at (1.5, 7.5).

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 = 6, c = 3.

    −2x2+6x+3=0-2x^2 + 6x + 3 = 0
  2. 2

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

    D=62−4⋅(−2)⋅3=36+24=60D = 6^2 - 4\cdot(-2)\cdot3 = 36 + 24 = 60
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−6±60−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{60}}{-4}
  5. 5

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

    x=3+152orx=3−152x = \frac{3 + \sqrt{15}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{15}}{2}
  6. 6

    As decimals: x ≈ -0.43649167 and x ≈ 3.4364917.

  7. 7

    Check numerically: for x ≈ -0.43649167 the left side is -4.4408921e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 6x + 3 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{15}}{2},\ \frac{3 - \sqrt{15}}{2} \]
\[ -2x^2 + 6x + 3 = 0 \]
\[ D = 6^2 - 4\cdot(-2)\cdot3 = 36 + 24 = 60 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{60}}{-4} \]
\[ x = \frac{3 + \sqrt{15}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{15}}{2} \]

The parabola y = -2x² + 6x + 3

xy
-1.5-10.5
-0.5-0.5
0.55.5
1.57.5
2.55.5
3.5-0.5
4.5-10.5

Questions

What are the solutions of -2x² + 6x + 3 = 0?

The solutions of -2x² + 6x + 3 = 0 are x = (3 + √15)/2 or x = (3 - √15)/2, that is x ≈ -0.43649167 and x ≈ 3.4364917.

Does -2x² + 6x + 3 = 0 factor?

Not over the integers, because the discriminant 60 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 6x + 3?

(3/2, 15/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -3/2.

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