If the product of the roots of the equation six 𝑥 squared plus two 𝑥 plus 𝑘 equals zero is negative four, what is the value of 𝑘?
So in this question, what we have is a quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. So when we have a quadratic in this form, what we actually have are a couple of relationships to deal with its roots. And that’s based around its coefficients. So first of all, the sum of the roots is equal to negative 𝑏 over 𝑎. And then we also know that the product of the roots is equal to 𝑐 over 𝑎.
Well, in this question, what we’re told is that the product of the roots is negative four. So therefore, it’s the second relationship we’re interested in because this is the relationship that tells us about the product of roots. And it tells us that this is equal to 𝑐 over 𝑎. So the first thing we’re gonna do is identify the 𝑎, 𝑏, and 𝑐 in our quadratic equation. And we can see that 𝑎 is equal to six, 𝑏 is equal to two, and 𝑐 is equal to 𝑘.
So therefore, using our second relationship, what we can say is that negative four, the product of our roots, is equal to 𝑘, our 𝑐-value, over our 𝑎-value, which is six. So now what we need to do is we want to work out the value of 𝑘. So we’re gonna multiply each side of our equation by six. So therefore, we’re gonna get negative 24 is equal to 𝑘. So what we can say that is if the product of the roots of equation six 𝑥 squared plus two 𝑥 plus 𝑘 equals zero is negative four, then the value of 𝑘 is negative 24.