WordGPT

Solve x² + 9x - 1 = 0

The solutions of x² + 9x - 1 = 0 are x = (-9 + √85)/2 or x = (-9 - √85)/2, that is x ≈ 0.10977223 and x ≈ -9.1097722.

x² +x += 0
x2+9x−1=0⇒x=−9+852, −9−852x^2 + 9x - 1 = 0 \Rightarrow x = \frac{-9 + \sqrt{85}}{2},\ \frac{-9 - \sqrt{85}}{2}
Discriminant D = 85 · x ≈ 0.109772, -9.10977

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

Explain it step by step in chatMake a short video of it
Graph of y = x² + 9x - 1-10-8-6-4-22-20-10102030x = 0.10977223x = -9.1097722vertexy = x² + 9x - 1
Graph of y = x² + 9x - 1. The curve crosses the x-axis at x = 0.10977223 and x = -9.1097722; the vertex is at (-4.5, -21.25).

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 = 9, c = -1.

    x2+9x−1=0x^2 + 9x - 1 = 0
  2. 2

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

    D=92−4⋅1⋅(−1)=81+4=85D = 9^2 - 4\cdot1\cdot(-1) = 81 + 4 = 85
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±852x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{85}}{2}
  5. 5

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

    x=−9+852orx=−9−852x = \frac{-9 + \sqrt{85}}{2}\quad\text{or}\quad x = \frac{-9 - \sqrt{85}}{2}
  6. 6

    As decimals: x ≈ 0.10977223 and x ≈ -9.1097722.

  7. 7

    Check numerically: for x ≈ 0.10977223 the left side is -1.110223e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 9x - 1 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{85}}{2},\ \frac{-9 - \sqrt{85}}{2} \]
\[ x^2 + 9x - 1 = 0 \]
\[ D = 9^2 - 4\cdot1\cdot(-1) = 81 + 4 = 85 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{85}}{2} \]
\[ x = \frac{-9 + \sqrt{85}}{2}\quad\text{or}\quad x = \frac{-9 - \sqrt{85}}{2} \]

The parabola y = x² + 9x - 1

xy
-7.5-12.25
-6.5-17.25
-5.5-20.25
-4.5-21.25
-3.5-20.25
-2.5-17.25
-1.5-12.25

Questions

What are the solutions of x² + 9x - 1 = 0?

The solutions of x² + 9x - 1 = 0 are x = (-9 + √85)/2 or x = (-9 - √85)/2, that is x ≈ 0.10977223 and x ≈ -9.1097722.

Does x² + 9x - 1 = 0 factor?

Not over the integers, because the discriminant 85 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 9x - 1?

(-9/2, -85/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9, product = c/a = -1.

Similar equations

    Solve x² + 9x - 1 = 0: x = (-9 + √85)/2 or x = (-9 - √85)/2 (steps) | WordGPT