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Solve x² + x - 10 = 0

The solutions of x² + x - 10 = 0 are x = (-1 + √41)/2 or x = (-1 - √41)/2, that is x ≈ 2.7015621 and x ≈ -3.7015621.

x² +x += 0
x2+x−10=0⇒x=−1+412, −1−412x^2 + x - 10 = 0 \Rightarrow x = \frac{-1 + \sqrt{41}}{2},\ \frac{-1 - \sqrt{41}}{2}
Discriminant D = 41 · x ≈ 2.70156, -3.70156

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

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Graph of y = x² + x - 10-6-4-224-10-55101520x = 2.7015621x = -3.7015621vertexy = x² + x - 10
Graph of y = x² + x - 10. The curve crosses the x-axis at x = 2.7015621 and x = -3.7015621; the vertex is at (-0.5, -10.25).

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 = 1, c = -10.

    x2+x−10=0x^2 + x - 10 = 0
  2. 2

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

    D=12−4⋅1⋅(−10)=1+40=41D = 1^2 - 4\cdot1\cdot(-10) = 1 + 40 = 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=−1±412x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{41}}{2}
  5. 5

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

    x=−1+412orx=−1−412x = \frac{-1 + \sqrt{41}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{41}}{2}
  6. 6

    As decimals: x ≈ 2.7015621 and x ≈ -3.7015621.

  7. 7

    Check numerically: for x ≈ 2.7015621 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + x - 10 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{41}}{2},\ \frac{-1 - \sqrt{41}}{2} \]
\[ x^2 + x - 10 = 0 \]
\[ D = 1^2 - 4\cdot1\cdot(-10) = 1 + 40 = 41 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{41}}{2} \]
\[ x = \frac{-1 + \sqrt{41}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{41}}{2} \]

The parabola y = x² + x - 10

xy
-3.5-1.25
-2.5-6.25
-1.5-9.25
-0.5-10.25
0.5-9.25
1.5-6.25
2.5-1.25

Questions

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

The solutions of x² + x - 10 = 0 are x = (-1 + √41)/2 or x = (-1 - √41)/2, that is x ≈ 2.7015621 and x ≈ -3.7015621.

Does x² + x - 10 = 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 = x² + x - 10?

(-1/2, -41/4). 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 = -10.

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    Solve x² + x - 10 = 0: x = (-1 + √41)/2 or x = (-1 - √41)/2 (steps) | WordGPT