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Solve x² + 7x - 1 = 0

The solutions of x² + 7x - 1 = 0 are x = (-7 + √53)/2 or x = (-7 - √53)/2, that is x ≈ 0.14005494 and x ≈ -7.1400549.

x² +x += 0
x2+7x−1=0⇒x=−7+532, −7−532x^2 + 7x - 1 = 0 \Rightarrow x = \frac{-7 + \sqrt{53}}{2},\ \frac{-7 - \sqrt{53}}{2}
Discriminant D = 53 · x ≈ 0.140055, -7.14005

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

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Graph of y = x² + 7x - 1-8-6-4-22-101020x = 0.14005494x = -7.1400549vertexy = x² + 7x - 1
Graph of y = x² + 7x - 1. The curve crosses the x-axis at x = 0.14005494 and x = -7.1400549; the vertex is at (-3.5, -13.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 7, c = -1.

    x2+7x−1=0x^2 + 7x - 1 = 0
  2. 2

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

    D=72−4⋅1⋅(−1)=49+4=53D = 7^2 - 4\cdot1\cdot(-1) = 49 + 4 = 53
  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±532x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{53}}{2}
  5. 5

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

    x=−7+532orx=−7−532x = \frac{-7 + \sqrt{53}}{2}\quad\text{or}\quad x = \frac{-7 - \sqrt{53}}{2}
  6. 6

    As decimals: x ≈ 0.14005494 and x ≈ -7.1400549.

  7. 7

    Check numerically: for x ≈ 0.14005494 the left side is -7.7715612e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 7x - 1 = 0\ \Rightarrow\ x = \frac{-7 + \sqrt{53}}{2},\ \frac{-7 - \sqrt{53}}{2} \]
\[ x^2 + 7x - 1 = 0 \]
\[ D = 7^2 - 4\cdot1\cdot(-1) = 49 + 4 = 53 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{53}}{2} \]
\[ x = \frac{-7 + \sqrt{53}}{2}\quad\text{or}\quad x = \frac{-7 - \sqrt{53}}{2} \]

The parabola y = x² + 7x - 1

xy
-6.5-4.25
-5.5-9.25
-4.5-12.25
-3.5-13.25
-2.5-12.25
-1.5-9.25
-0.5-4.25

Questions

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

The solutions of x² + 7x - 1 = 0 are x = (-7 + √53)/2 or x = (-7 - √53)/2, that is x ≈ 0.14005494 and x ≈ -7.1400549.

Does x² + 7x - 1 = 0 factor?

Not over the integers, because the discriminant 53 is not a perfect square. Use the quadratic formula.

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

(-7/2, -53/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7, product = c/a = -1.

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    Solve x² + 7x - 1 = 0: x = (-7 + √53)/2 or x = (-7 - √53)/2 (steps) | WordGPT