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Solve x² - 9x - 3 = 0

The solutions of x² - 9x - 3 = 0 are x = (9 + √93)/2 or x = (9 - √93)/2, that is x ≈ 9.3218254 and x ≈ -0.32182538.

x² +x += 0
x2−9x−3=0⇒x=9+932, 9−932x^2 - 9x - 3 = 0 \Rightarrow x = \frac{9 + \sqrt{93}}{2},\ \frac{9 - \sqrt{93}}{2}
Discriminant D = 93 · x ≈ 9.32183, -0.321825

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

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Graph of y = x² - 9x - 3-2246810-30-20-10102030x = 9.3218254x = -0.32182538vertexy = x² - 9x - 3
Graph of y = x² - 9x - 3. The curve crosses the x-axis at x = 9.3218254 and x = -0.32182538; the vertex is at (4.5, -23.25).

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 = -3.

    x2−9x−3=0x^2 - 9x - 3 = 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⋅(−3)=81+12=93D = (-9)^2 - 4\cdot1\cdot(-3) = 81 + 12 = 93
  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±932x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{93}}{2}
  5. 5

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

    x=9+932orx=9−932x = \frac{9 + \sqrt{93}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{93}}{2}
  6. 6

    As decimals: x ≈ 9.3218254 and x ≈ -0.32182538.

  7. 7

    Check numerically: for x ≈ 9.3218254 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 9x - 3 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{93}}{2},\ \frac{9 - \sqrt{93}}{2} \]
\[ x^2 - 9x - 3 = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot(-3) = 81 + 12 = 93 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{93}}{2} \]
\[ x = \frac{9 + \sqrt{93}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{93}}{2} \]

The parabola y = x² - 9x - 3

xy
1.5-14.25
2.5-19.25
3.5-22.25
4.5-23.25
5.5-22.25
6.5-19.25
7.5-14.25

Questions

What are the solutions of x² - 9x - 3 = 0?

The solutions of x² - 9x - 3 = 0 are x = (9 + √93)/2 or x = (9 - √93)/2, that is x ≈ 9.3218254 and x ≈ -0.32182538.

Does x² - 9x - 3 = 0 factor?

Not over the integers, because the discriminant 93 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 9x - 3?

(9/2, -93/4). 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 = -3.

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    Solve x² - 9x - 3 = 0: x = (9 + √93)/2 or x = (9 - √93)/2 (steps) | WordGPT