2х^2+14х+24

To analyze the quadratic expression \(2x^2 + 14x + 24\), we can do a few things: factor it, complete the square, or find the roots.

Factoring

1. First, we can factor out the greatest common factor (GCF). In this case, the GCF is 2:

   \[ 2(x^2 + 7x + 12) \]

2. Now, we can factor the quadratic \(x^2 + 7x + 12\). We need two numbers that multiply to 12 and add up to 7. The numbers 3 and 4 fit this criteria:

   \[ x^2 + 7x + 12 = (x + 3)(x + 4) \]

3. Putting it all together, the factored form of the original expression is:

   \[ 2(x + 3)(x + 4) \]

Finding the Roots

To find the roots of the equation \(2x^2 + 14x + 24 = 0\):

1. Set the factored form equal to zero:

   \[ 2(x + 3)(x + 4) = 0 \]

2. This gives us two equations to solve:

   \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \] \[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

So the roots are \(x = -3\) and \(x = -4\).

Summary

The expression \(2x^2 + 14x + 24\) factors to \(2(x + 3)(x + 4)\), and its roots are \(x = -3\) and \(x = -4\).