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Solve 2x² + 10x - 1 = 0

The solutions of 2x² + 10x - 1 = 0 are x = (-5 + 3√3)/2 or x = (-5 - 3√3)/2, that is x ≈ 0.098076211 and x ≈ -5.0980762.

x² +x += 0
2x2+10x−1=0⇒x=−5+332, −5−3322x^2 + 10x - 1 = 0 \Rightarrow x = \frac{-5 + 3\sqrt{3}}{2},\ \frac{-5 - 3\sqrt{3}}{2}
Discriminant D = 108 · x ≈ 0.0980762, -5.09808

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

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Graph of y = 2x² + 10x - 1-7-6-5-4-3-2-112-20-1010203040x = 0.098076211x = -5.0980762vertexy = 2x² + 10x - 1
Graph of y = 2x² + 10x - 1. The curve crosses the x-axis at x = 0.098076211 and x = -5.0980762; the vertex is at (-2.5, -13.5).

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 = 10, c = -1.

    2x2+10x−1=02x^2 + 10x - 1 = 0
  2. 2

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

    D=102−4⋅2⋅(−1)=100+8=108D = 10^2 - 4\cdot2\cdot(-1) = 100 + 8 = 108
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±1084x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{108}}{4}
  5. 5

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

    x=−5+332orx=−5−332x = \frac{-5 + 3\sqrt{3}}{2}\quad\text{or}\quad x = \frac{-5 - 3\sqrt{3}}{2}
  6. 6

    As decimals: x ≈ 0.098076211 and x ≈ -5.0980762.

  7. 7

    Check numerically: for x ≈ 0.098076211 the left side is 6.6613381e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 10x - 1 = 0\ \Rightarrow\ x = \frac{-5 + 3\sqrt{3}}{2},\ \frac{-5 - 3\sqrt{3}}{2} \]
\[ 2x^2 + 10x - 1 = 0 \]
\[ D = 10^2 - 4\cdot2\cdot(-1) = 100 + 8 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{108}}{4} \]
\[ x = \frac{-5 + 3\sqrt{3}}{2}\quad\text{or}\quad x = \frac{-5 - 3\sqrt{3}}{2} \]

The parabola y = 2x² + 10x - 1

xy
-5.54.5
-4.5-5.5
-3.5-11.5
-2.5-13.5
-1.5-11.5
-0.5-5.5
0.54.5

Questions

What are the solutions of 2x² + 10x - 1 = 0?

The solutions of 2x² + 10x - 1 = 0 are x = (-5 + 3√3)/2 or x = (-5 - 3√3)/2, that is x ≈ 0.098076211 and x ≈ -5.0980762.

Does 2x² + 10x - 1 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 10x - 1?

(-5/2, -27/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = -1/2.

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    Solve 2x² + 10x - 1 = 0: x = (-5 + 3√3)/2 or x = (-5 - 3√3)/2 (steps) | WordGPT