WordGPT

Solve 3x² - 9x - 6 = 0

The solutions of 3x² - 9x - 6 = 0 are x = (3 + √17)/2 or x = (3 - √17)/2, that is x ≈ 3.5615528 and x ≈ -0.56155281.

x² +x += 0
3x2−9x−6=0⇒x=3+172, 3−1723x^2 - 9x - 6 = 0 \Rightarrow x = \frac{3 + \sqrt{17}}{2},\ \frac{3 - \sqrt{17}}{2}
Discriminant D = 153 · x ≈ 3.56155, -0.561553

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 = 3x² - 9x - 6-3-2-1123456-20-101020304050x = 3.5615528x = -0.56155281vertexy = 3x² - 9x - 6
Graph of y = 3x² - 9x - 6. The curve crosses the x-axis at x = 3.5615528 and x = -0.56155281; the vertex is at (1.5, -12.75).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = -9, c = -6.

    3x2−9x−6=03x^2 - 9x - 6 = 0
  2. 2

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

    D=(−9)2−4⋅3⋅(−6)=81+72=153D = (-9)^2 - 4\cdot3\cdot(-6) = 81 + 72 = 153
  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±1536x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{153}}{6}
  5. 5

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

    x=3+172orx=3−172x = \frac{3 + \sqrt{17}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{17}}{2}
  6. 6

    As decimals: x ≈ 3.5615528 and x ≈ -0.56155281.

  7. 7

    Check numerically: for x ≈ 3.5615528 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 9x - 6 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{17}}{2},\ \frac{3 - \sqrt{17}}{2} \]
\[ 3x^2 - 9x - 6 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot(-6) = 81 + 72 = 153 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{153}}{6} \]
\[ x = \frac{3 + \sqrt{17}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{17}}{2} \]

The parabola y = 3x² - 9x - 6

xy
-1.514.25
-0.5-0.75
0.5-9.75
1.5-12.75
2.5-9.75
3.5-0.75
4.514.25

Questions

What are the solutions of 3x² - 9x - 6 = 0?

The solutions of 3x² - 9x - 6 = 0 are x = (3 + √17)/2 or x = (3 - √17)/2, that is x ≈ 3.5615528 and x ≈ -0.56155281.

Does 3x² - 9x - 6 = 0 factor?

Not over the integers, because the discriminant 153 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 9x - 6?

(3/2, -51/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -2.

Similar equations

    Solve 3x² - 9x - 6 = 0: x = (3 + √17)/2 or x = (3 - √17)/2 (steps) | WordGPT