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Solve x² + 9x + 5 = 0

The solutions of x² + 9x + 5 = 0 are x = (-9 + √61)/2 or x = (-9 - √61)/2, that is x ≈ -0.59487516 and x ≈ -8.4051248.

x² +x += 0
x2+9x+5=0⇒x=−9+612, −9−612x^2 + 9x + 5 = 0 \Rightarrow x = \frac{-9 + \sqrt{61}}{2},\ \frac{-9 - \sqrt{61}}{2}
Discriminant D = 61 · x ≈ -0.594875, -8.40512

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

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Graph of y = x² + 9x + 5-10-8-6-4-2-20-101020x = -0.59487516x = -8.4051248vertexy = x² + 9x + 5
Graph of y = x² + 9x + 5. The curve crosses the x-axis at x = -0.59487516 and x = -8.4051248; the vertex is at (-4.5, -15.25).

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 = 5.

    x2+9x+5=0x^2 + 9x + 5 = 0
  2. 2

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

    D=92−4⋅1⋅5=81−20=61D = 9^2 - 4\cdot1\cdot5 = 81 - 20 = 61
  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±612x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{61}}{2}
  5. 5

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

    x=−9+612orx=−9−612x = \frac{-9 + \sqrt{61}}{2}\quad\text{or}\quad x = \frac{-9 - \sqrt{61}}{2}
  6. 6

    As decimals: x ≈ -0.59487516 and x ≈ -8.4051248.

  7. 7

    Check numerically: for x ≈ -0.59487516 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 9x + 5 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{61}}{2},\ \frac{-9 - \sqrt{61}}{2} \]
\[ x^2 + 9x + 5 = 0 \]
\[ D = 9^2 - 4\cdot1\cdot5 = 81 - 20 = 61 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{61}}{2} \]
\[ x = \frac{-9 + \sqrt{61}}{2}\quad\text{or}\quad x = \frac{-9 - \sqrt{61}}{2} \]

The parabola y = x² + 9x + 5

xy
-7.5-6.25
-6.5-11.25
-5.5-14.25
-4.5-15.25
-3.5-14.25
-2.5-11.25
-1.5-6.25

Questions

What are the solutions of x² + 9x + 5 = 0?

The solutions of x² + 9x + 5 = 0 are x = (-9 + √61)/2 or x = (-9 - √61)/2, that is x ≈ -0.59487516 and x ≈ -8.4051248.

Does x² + 9x + 5 = 0 factor?

Not over the integers, because the discriminant 61 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 9x + 5?

(-9/2, -61/4). 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 = 5.

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    Solve x² + 9x + 5 = 0: x = (-9 + √61)/2 or x = (-9 - √61)/2 (steps) | WordGPT