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Solve -x² + 2x + 9 = 0

The solutions of -x² + 2x + 9 = 0 are x = 1 + √10 or x = 1 - √10, that is x ≈ -2.1622777 and x ≈ 4.1622777.

x² +x += 0
−x2+2x+9=0⇒x=1+10, 1−10-x^2 + 2x + 9 = 0 \Rightarrow x = 1 + \sqrt{10},\ 1 - \sqrt{10}
Discriminant D = 40 · x ≈ -2.16228, 4.16228

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

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Graph of y = -x² + 2x + 9-4-2246-20-15-10-5510x = -2.1622777x = 4.1622777vertexy = -x² + 2x + 9
Graph of y = -x² + 2x + 9. The curve crosses the x-axis at x = -2.1622777 and x = 4.1622777; the vertex is at (1, 10).

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 = 2, c = 9.

    −x2+2x+9=0-x^2 + 2x + 9 = 0
  2. 2

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

    D=22−4⋅(−1)⋅9=4+36=40D = 2^2 - 4\cdot(-1)\cdot9 = 4 + 36 = 40
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−2±40−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{40}}{-2}
  5. 5

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

    x=1+10orx=1−10x = 1 + \sqrt{10}\quad\text{or}\quad x = 1 - \sqrt{10}
  6. 6

    As decimals: x ≈ -2.1622777 and x ≈ 4.1622777.

  7. 7

    Check numerically: for x ≈ -2.1622777 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 2x + 9 = 0\ \Rightarrow\ x = 1 + \sqrt{10},\ 1 - \sqrt{10} \]
\[ -x^2 + 2x + 9 = 0 \]
\[ D = 2^2 - 4\cdot(-1)\cdot9 = 4 + 36 = 40 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{40}}{-2} \]
\[ x = 1 + \sqrt{10}\quad\text{or}\quad x = 1 - \sqrt{10} \]

The parabola y = -x² + 2x + 9

xy
-21
-16
09
110
29
36
41

Questions

What are the solutions of -x² + 2x + 9 = 0?

The solutions of -x² + 2x + 9 = 0 are x = 1 + √10 or x = 1 - √10, that is x ≈ -2.1622777 and x ≈ 4.1622777.

Does -x² + 2x + 9 = 0 factor?

Not over the integers, because the discriminant 40 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 2x + 9?

(1, 10). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = -9.

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    Solve -x² + 2x + 9 = 0: x = 1 + √10 or x = 1 - √10 (steps) | WordGPT