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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 ≈ 8.4051248 and x ≈ 0.59487516.

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 ≈ 8.40512, 0.594875

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

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Graph of y = x² - 9x + 5246810-20-101020x = 8.4051248x = 0.59487516vertexy = x² - 9x + 5
Graph of y = x² - 9x + 5. The curve crosses the x-axis at x = 8.4051248 and x = 0.59487516; 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=(−9)2−4⋅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 ≈ 8.4051248 and x ≈ 0.59487516.

  7. 7

    Check numerically: for x ≈ 8.4051248 the left side is 0 (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
1.5-6.25
2.5-11.25
3.5-14.25
4.5-15.25
5.5-14.25
6.5-11.25
7.5-6.25

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 ≈ 8.4051248 and x ≈ 0.59487516.

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