Questions: Factor completely. -3v^2+14v-8

Solution Steps

To factor the quadratic expression \(-3v^2 + 14v - 8\), we first look for two numbers that multiply to the product of the leading coefficient and the constant term (i.e., \(-3 \times -8 = 24\)) and add up to the middle coefficient (i.e., \(14\)). Once these numbers are found, we can use them to split the middle term and factor by grouping.

We start with the quadratic expression: \[ -3v^2 + 14v - 8 \]

To factor the expression, we first rewrite it in a more manageable form. We can factor out a negative sign: \[ -(3v^2 - 14v + 8) \] Next, we need to factor the quadratic \(3v^2 - 14v + 8\). We look for two numbers that multiply to \(3 \times 8 = 24\) and add to \(-14\). The numbers \(-12\) and \(-2\) satisfy these conditions.

We can rewrite the quadratic as: \[ 3v^2 - 12v - 2v + 8 \] Now, we group the terms: \[ (3v^2 - 12v) + (-2v + 8) \] Factoring each group gives us: \[ 3v(v - 4) - 2(v - 4) \] Now, we can factor out the common term \((v - 4)\): \[ (3v - 2)(v - 4) \]

Including the negative sign we factored out earlier, we have: \[ -(v - 4)(3v - 2) \]

Final Answer