6x^2+x-2

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:

1. \(3x + 2 = 0\) ⟹ \(3x = -2\) ⟹ \(x = -\frac{2}{3}\)
2. \(2x - 1 = 0\) ⟹ \(2x = 1\) ⟹ \(x = \frac{1}{2}\)

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!