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Solve 2x² + 6x - 4 = 0

The solutions of 2x² + 6x - 4 = 0 are x = (-3 + √17)/2 or x = (-3 - √17)/2, that is x ≈ 0.56155281 and x ≈ -3.5615528.

x² +x += 0
2x2+6x−4=0⇒x=−3+172, −3−1722x^2 + 6x - 4 = 0 \Rightarrow x = \frac{-3 + \sqrt{17}}{2},\ \frac{-3 - \sqrt{17}}{2}
Discriminant D = 68 · x ≈ 0.561553, -3.56155

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

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Graph of y = 2x² + 6x - 4-6-5-4-3-2-1123-10102030x = 0.56155281x = -3.5615528vertexy = 2x² + 6x - 4
Graph of y = 2x² + 6x - 4. The curve crosses the x-axis at x = 0.56155281 and x = -3.5615528; the vertex is at (-1.5, -8.5).

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 = -4.

    2x2+6x−4=02x^2 + 6x - 4 = 0
  2. 2

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

    D=62−4⋅2⋅(−4)=36+32=68D = 6^2 - 4\cdot2\cdot(-4) = 36 + 32 = 68
  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±684x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{68}}{4}
  5. 5

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

    x=−3+172orx=−3−172x = \frac{-3 + \sqrt{17}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{17}}{2}
  6. 6

    As decimals: x ≈ 0.56155281 and x ≈ -3.5615528.

  7. 7

    Check numerically: for x ≈ 0.56155281 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 6x - 4 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{17}}{2},\ \frac{-3 - \sqrt{17}}{2} \]
\[ 2x^2 + 6x - 4 = 0 \]
\[ D = 6^2 - 4\cdot2\cdot(-4) = 36 + 32 = 68 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{68}}{4} \]
\[ x = \frac{-3 + \sqrt{17}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{17}}{2} \]

The parabola y = 2x² + 6x - 4

xy
-4.59.5
-3.5-0.5
-2.5-6.5
-1.5-8.5
-0.5-6.5
0.5-0.5
1.59.5

Questions

What are the solutions of 2x² + 6x - 4 = 0?

The solutions of 2x² + 6x - 4 = 0 are x = (-3 + √17)/2 or x = (-3 - √17)/2, that is x ≈ 0.56155281 and x ≈ -3.5615528.

Does 2x² + 6x - 4 = 0 factor?

Not over the integers, because the discriminant 68 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 6x - 4?

(-3/2, -17/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -2.

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    Solve 2x² + 6x - 4 = 0: x = (-3 + √17)/2 or x = (-3 - √17)/2 (steps) | WordGPT