To work with the quadratic expression (6x^2 + x - 2), you may be looking to factor it, find its roots, or analyze it in some way. Here are some steps depending on what you need:
Factoring
To factor the quadratic expression (6x^2 + x - 2), we can look for two numbers that multiply to (6 \times -2 = -12) and add to (1) (the coefficient of (x)).
The numbers that satisfy these conditions are (4) and (-3).
Now we can rewrite the middle term:
[
6x^2 + 4x - 3x - 2
]
Next, we can group the terms:
[
(6x^2 + 4x) + (-3x - 2)
]
Factor each group:
[
2x(3x + 2) - 1(3x + 2)
]
Now, factor out the common factor ((3x + 2)):
[
(3x + 2)(2x - 1)
]
Finding Roots
To find the roots of the equation (6x^2 + x - 2 = 0), we can use the factored form:
[
(3x + 2)(2x - 1) = 0
]
Setting each factor to zero gives:
- (3x + 2 = 0) ⟹ (3x = -2) ⟹ (x = -\frac{2}{3})
- (2x - 1 = 0) ⟹ (2x = 1) ⟹ (x = \frac{1}{2})
So, the roots are (x = -\frac{2}{3}) and (x = \frac{1}{2}).
Summary
- The factored form of (6x^2 + x - 2) is ((3x + 2)(2x - 1)).
- The roots are (x = -\frac{2}{3}) and (x = \frac{1}{2}).
If you need further assistance or specific calculations, feel free to ask!
