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Solve -2x² - 7x - 1 = 0

The solutions of -2x² - 7x - 1 = 0 are x = (-7 + √41)/4 or x = (-7 - √41)/4, that is x ≈ -3.3507811 and x ≈ -0.14921894.

x² +x += 0
−2x2−7x−1=0⇒x=−7+414, −7−414-2x^2 - 7x - 1 = 0 \Rightarrow x = \frac{-7 + \sqrt{41}}{4},\ \frac{-7 - \sqrt{41}}{4}
Discriminant D = 41 · x ≈ -3.35078, -0.149219

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

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Graph of y = -2x² - 7x - 1-5-4-3-2-112-30-25-20-15-10-55x = -3.3507811x = -0.14921894vertexy = -2x² - 7x - 1
Graph of y = -2x² - 7x - 1. The curve crosses the x-axis at x = -3.3507811 and x = -0.14921894; the vertex is at (-1.75, 5.125).

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 = -7, c = -1.

    −2x2−7x−1=0-2x^2 - 7x - 1 = 0
  2. 2

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

    D=(−7)2−4⋅(−2)⋅(−1)=49−8=41D = (-7)^2 - 4\cdot(-2)\cdot(-1) = 49 - 8 = 41
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=7±41−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{41}}{-4}
  5. 5

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

    x=−7+414orx=−7−414x = \frac{-7 + \sqrt{41}}{4}\quad\text{or}\quad x = \frac{-7 - \sqrt{41}}{4}
  6. 6

    As decimals: x ≈ -3.3507811 and x ≈ -0.14921894.

  7. 7

    Check numerically: for x ≈ -3.3507811 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 7x - 1 = 0\ \Rightarrow\ x = \frac{-7 + \sqrt{41}}{4},\ \frac{-7 - \sqrt{41}}{4} \]
\[ -2x^2 - 7x - 1 = 0 \]
\[ D = (-7)^2 - 4\cdot(-2)\cdot(-1) = 49 - 8 = 41 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{41}}{-4} \]
\[ x = \frac{-7 + \sqrt{41}}{4}\quad\text{or}\quad x = \frac{-7 - \sqrt{41}}{4} \]

The parabola y = -2x² - 7x - 1

xy
-4.75-12.875
-3.75-2.875
-2.753.125
-1.755.125
-0.753.125
0.25-2.875
1.25-12.875

Questions

What are the solutions of -2x² - 7x - 1 = 0?

The solutions of -2x² - 7x - 1 = 0 are x = (-7 + √41)/4 or x = (-7 - √41)/4, that is x ≈ -3.3507811 and x ≈ -0.14921894.

Does -2x² - 7x - 1 = 0 factor?

Not over the integers, because the discriminant 41 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 7x - 1?

(-7/4, 41/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -7/2, product = c/a = 1/2.

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    Solve -2x² - 7x - 1 = 0: x = (-7 + √41)/4 or x = (-7 - √41)/4 (steps) | WordGPT