WordGPT

Solve 3x² + 7x - 11 = 0

The solutions of 3x² + 7x - 11 = 0 are x = (-7 + √181)/6 or x = (-7 - √181)/6, that is x ≈ 1.075604 and x ≈ -3.4089373.

x² +x += 0
3x2+7x−11=0⇒x=−7+1816, −7−18163x^2 + 7x - 11 = 0 \Rightarrow x = \frac{-7 + \sqrt{181}}{6},\ \frac{-7 - \sqrt{181}}{6}
Discriminant D = 181 · x ≈ 1.0756, -3.40894

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 + 7x - 11-5-4-3-2-1123-20-101020304050x = 1.075604x = -3.4089373vertexy = 3x² + 7x - 11
Graph of y = 3x² + 7x - 11. The curve crosses the x-axis at x = 1.075604 and x = -3.4089373; the vertex is at (-1.1666667, -15.083333).

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 = 7, c = -11.

    3x2+7x−11=03x^2 + 7x - 11 = 0
  2. 2

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

    D=72−4⋅3⋅(−11)=49+132=181D = 7^2 - 4\cdot3\cdot(-11) = 49 + 132 = 181
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−7±1816x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{181}}{6}
  5. 5

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

    x=−7+1816orx=−7−1816x = \frac{-7 + \sqrt{181}}{6}\quad\text{or}\quad x = \frac{-7 - \sqrt{181}}{6}
  6. 6

    As decimals: x ≈ 1.075604 and x ≈ -3.4089373.

  7. 7

    Check numerically: for x ≈ 1.075604 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 7x - 11 = 0\ \Rightarrow\ x = \frac{-7 + \sqrt{181}}{6},\ \frac{-7 - \sqrt{181}}{6} \]
\[ 3x^2 + 7x - 11 = 0 \]
\[ D = 7^2 - 4\cdot3\cdot(-11) = 49 + 132 = 181 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{181}}{6} \]
\[ x = \frac{-7 + \sqrt{181}}{6}\quad\text{or}\quad x = \frac{-7 - \sqrt{181}}{6} \]

The parabola y = 3x² + 7x - 11

xy
-4.166711.917267
-3.1667-3.0829333
-2.1667-12.083133
-1.1667-15.083333
-0.1667-12.083533
0.8333-3.0837333
1.833311.916067

Questions

What are the solutions of 3x² + 7x - 11 = 0?

The solutions of 3x² + 7x - 11 = 0 are x = (-7 + √181)/6 or x = (-7 - √181)/6, that is x ≈ 1.075604 and x ≈ -3.4089373.

Does 3x² + 7x - 11 = 0 factor?

Not over the integers, because the discriminant 181 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 7x - 11?

(-7/6, -181/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7/3, product = c/a = -11/3.

Similar equations

    Solve 3x² + 7x - 11 = 0: x = (-7 + √181)/6 or x = (-7 - √181)/6 (steps) | WordGPT