WordGPT

Solve 3x² + 8x - 11 = 0

The solutions of 3x² + 8x - 11 = 0 are x = 1 or x = -11/3.

x² +x += 0
3x2+8x−11=0⇒x=1, −1133x^2 + 8x - 11 = 0 \Rightarrow x = 1,\ -\frac{11}{3}
Discriminant D = 196 · x ≈ 1, -3.66667

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 + 8x - 11-6-5-4-3-2-1123-20-101020304050x = 1x = -3.6666667vertexy = 3x² + 8x - 11
Graph of y = 3x² + 8x - 11. The curve crosses the x-axis at x = 1 and x = -3.6666667; the vertex is at (-1.3333333, -16.333333).

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 = 8, c = -11.

    3x2+8x−11=03x^2 + 8x - 11 = 0
  2. 2

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

    D=82−4⋅3⋅(−11)=64+132=196D = 8^2 - 4\cdot3\cdot(-11) = 64 + 132 = 196
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−1)(3x+11)=0\left(x - 1\right)\left(3x + 11\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−1=0⇒x=13x+11=0⇒x=−113x - 1 = 0 \Rightarrow x = 1\qquad 3x + 11 = 0 \Rightarrow x = -\frac{11}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±1966x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{196}}{6}
  7. 7

    Simplify to get the solutions:

    x=1orx=−113x = 1\quad\text{or}\quad x = -\frac{11}{3}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 3·(1) + 8·(1) + (-11) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 8x - 11 = 0\ \Rightarrow\ x = 1,\ -\frac{11}{3} \]
\[ 3x^2 + 8x - 11 = 0 \]
\[ D = 8^2 - 4\cdot3\cdot(-11) = 64 + 132 = 196 \]
\[ \left(x - 1\right)\left(3x + 11\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x + 11 = 0 \Rightarrow x = -\frac{11}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{196}}{6} \]
\[ x = 1\quad\text{or}\quad x = -\frac{11}{3} \]

The parabola y = 3x² + 8x - 11

xy
-4.333310.666067
-3.3333-4.3337333
-2.3333-13.333533
-1.3333-16.333333
-0.3333-13.333133
0.6667-4.3329333
1.666710.667267

Questions

What are the solutions of 3x² + 8x - 11 = 0?

The solutions of 3x² + 8x - 11 = 0 are x = 1 or x = -11/3.

Does 3x² + 8x - 11 = 0 factor?

Yes: (x-1)(3x+11) = 0.

What is the vertex of y = 3x² + 8x - 11?

(-4/3, -49/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8/3, product = c/a = -11/3.

Similar equations

    Solve 3x² + 8x - 11 = 0: x = 1 or x = -11/3 (steps) | WordGPT