WordGPT

Solve 3x² - 5x - 11 = 0

The solutions of 3x² - 5x - 11 = 0 are x = (5 + √157)/6 or x = (5 - √157)/6, that is x ≈ 2.9216607 and x ≈ -1.254994.

x² +x += 0
3x2−5x−11=0⇒x=5+1576, 5−15763x^2 - 5x - 11 = 0 \Rightarrow x = \frac{5 + \sqrt{157}}{6},\ \frac{5 - \sqrt{157}}{6}
Discriminant D = 157 · x ≈ 2.92166, -1.25499

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 - 5x - 11-3-2-112345-20-101020304050x = 2.9216607x = -1.254994vertexy = 3x² - 5x - 11
Graph of y = 3x² - 5x - 11. The curve crosses the x-axis at x = 2.9216607 and x = -1.254994; the vertex is at (0.83333333, -13.083333).

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 = -5, c = -11.

    3x2−5x−11=03x^2 - 5x - 11 = 0
  2. 2

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

    D=(−5)2−4⋅3⋅(−11)=25+132=157D = (-5)^2 - 4\cdot3\cdot(-11) = 25 + 132 = 157
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=5±1576x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{157}}{6}
  5. 5

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

    x=5+1576orx=5−1576x = \frac{5 + \sqrt{157}}{6}\quad\text{or}\quad x = \frac{5 - \sqrt{157}}{6}
  6. 6

    As decimals: x ≈ 2.9216607 and x ≈ -1.254994.

  7. 7

    Check numerically: for x ≈ 2.9216607 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 5x - 11 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{157}}{6},\ \frac{5 - \sqrt{157}}{6} \]
\[ 3x^2 - 5x - 11 = 0 \]
\[ D = (-5)^2 - 4\cdot3\cdot(-11) = 25 + 132 = 157 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{157}}{6} \]
\[ x = \frac{5 + \sqrt{157}}{6}\quad\text{or}\quad x = \frac{5 - \sqrt{157}}{6} \]

The parabola y = 3x² - 5x - 11

xy
-2.166713.917267
-1.1667-1.0829333
-0.1667-10.083133
0.8333-13.083333
1.8333-10.083533
2.8333-1.0837333
3.833313.916067

Questions

What are the solutions of 3x² - 5x - 11 = 0?

The solutions of 3x² - 5x - 11 = 0 are x = (5 + √157)/6 or x = (5 - √157)/6, that is x ≈ 2.9216607 and x ≈ -1.254994.

Does 3x² - 5x - 11 = 0 factor?

Not over the integers, because the discriminant 157 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 5x - 11?

(5/6, -157/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/3, product = c/a = -11/3.

Similar equations

    Solve 3x² - 5x - 11 = 0: x = (5 + √157)/6 or x = (5 - √157)/6 (steps) | WordGPT