WordGPT

Solve 3x² + 3x - 6 = 0

The solutions of 3x² + 3x - 6 = 0 are x = 1 or x = -2.

x² +x += 0
3x2+3x−6=0⇒x=1, −23x^2 + 3x - 6 = 0 \Rightarrow x = 1,\ -2
Discriminant D = 81 · x ≈ 1, -2

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 + 3x - 6-4-3-2-1123-1010203040x = 1x = -2vertexy = 3x² + 3x - 6
Graph of y = 3x² + 3x - 6. The curve crosses the x-axis at x = 1 and x = -2; the vertex is at (-0.5, -6.75).

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 = 3, c = -6.

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

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

    D=32−4⋅3⋅(−6)=9+72=81D = 3^2 - 4\cdot3\cdot(-6) = 9 + 72 = 81
  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:

    3(x−1)(x+2)=03\left(x - 1\right)\left(x + 2\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=1x+2=0⇒x=−2x - 1 = 0 \Rightarrow x = 1\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−3±816x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{81}}{6}
  7. 7

    Simplify to get the solutions:

    x=1orx=−2x = 1\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 3·(1) + 3·(1) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 3x - 6 = 0\ \Rightarrow\ x = 1,\ -2 \]
\[ 3x^2 + 3x - 6 = 0 \]
\[ D = 3^2 - 4\cdot3\cdot(-6) = 9 + 72 = 81 \]
\[ 3\left(x - 1\right)\left(x + 2\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{81}}{6} \]
\[ x = 1\quad\text{or}\quad x = -2 \]

The parabola y = 3x² + 3x - 6

xy
-3.520.25
-2.55.25
-1.5-3.75
-0.5-6.75
0.5-3.75
1.55.25
2.520.25

Questions

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

The solutions of 3x² + 3x - 6 = 0 are x = 1 or x = -2.

Does 3x² + 3x - 6 = 0 factor?

Yes: 3(x-1)(x+2) = 0.

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

(-1/2, -27/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = -2.

Similar equations

    Solve 3x² + 3x - 6 = 0: x = 1 or x = -2 (steps) | WordGPT