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Solve 2x² + 2x - 6 = 0

The solutions of 2x² + 2x - 6 = 0 are x = (-1 + √13)/2 or x = (-1 - √13)/2, that is x ≈ 1.3027756 and x ≈ -2.3027756.

x² +x += 0
2x2+2x−6=0⇒x=−1+132, −1−1322x^2 + 2x - 6 = 0 \Rightarrow x = \frac{-1 + \sqrt{13}}{2},\ \frac{-1 - \sqrt{13}}{2}
Discriminant D = 52 · x ≈ 1.30278, -2.30278

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

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Graph of y = 2x² + 2x - 6-4-3-2-1123-10102030x = 1.3027756x = -2.3027756vertexy = 2x² + 2x - 6
Graph of y = 2x² + 2x - 6. The curve crosses the x-axis at x = 1.3027756 and x = -2.3027756; the vertex is at (-0.5, -6.5).

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 = 2, c = -6.

    2x2+2x−6=02x^2 + 2x - 6 = 0
  2. 2

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

    D=22−4⋅2⋅(−6)=4+48=52D = 2^2 - 4\cdot2\cdot(-6) = 4 + 48 = 52
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−2±524x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{52}}{4}
  5. 5

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

    x=−1+132orx=−1−132x = \frac{-1 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{13}}{2}
  6. 6

    As decimals: x ≈ 1.3027756 and x ≈ -2.3027756.

  7. 7

    Check numerically: for x ≈ 1.3027756 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 2x - 6 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{13}}{2},\ \frac{-1 - \sqrt{13}}{2} \]
\[ 2x^2 + 2x - 6 = 0 \]
\[ D = 2^2 - 4\cdot2\cdot(-6) = 4 + 48 = 52 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{52}}{4} \]
\[ x = \frac{-1 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{13}}{2} \]

The parabola y = 2x² + 2x - 6

xy
-3.511.5
-2.51.5
-1.5-4.5
-0.5-6.5
0.5-4.5
1.51.5
2.511.5

Questions

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

The solutions of 2x² + 2x - 6 = 0 are x = (-1 + √13)/2 or x = (-1 - √13)/2, that is x ≈ 1.3027756 and x ≈ -2.3027756.

Does 2x² + 2x - 6 = 0 factor?

Not over the integers, because the discriminant 52 is not a perfect square. Use the quadratic formula.

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

(-1/2, -13/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = -3.

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    Solve 2x² + 2x - 6 = 0: x = (-1 + √13)/2 or x = (-1 - √13)/2 (steps) | WordGPT