Video Transcript
What are the solutions of the quadratic equation six 𝑥 squared minus 18𝑥 minus 24 equals zero?
If we start with six 𝑥 squared minus 18𝑥 minus 24 equals zero, we see that 6, 18, and 24 are all divisible by six. And so, we can divide this entire equation by six to reduce. After that, six 𝑥 squared divided by six equals 𝑥 squared. 18 𝑥 divided by six equals three 𝑥. Be careful with your signs; we’re subtracting here. Bring that down. 24 divided by six equals four. And again, be careful to bring down that subtraction. Zero divided by six is zero.
We now have 𝑥 squared minus three 𝑥 minus four equals zero. And this equation will be a little bit easier to factor. The solutions are the 𝑥-values that make this statement true. And we want to factor this equation. We know that 𝑥 times 𝑥 equals 𝑥 squared. So, the first term of each factor is just going to be 𝑥. Our second terms have to multiply together to equal negative four and have to add together to equal negative three.
This means we need factors of negative four. If we multiply one by negative four, we get negative four. If we multiply negative one by positive four, we get negative four. And if we multiply two by negative two, we get negative four. And remember, we’re looking for the factor pair which sums to negative three. If we add together each of these factor pairs, only one of them will add to equal negative three. This means that the terms we’re looking for are one and negative four.
And so, we plug those in. And we have two factors, 𝑥 plus one times 𝑥 minus four is equal to zero. And then, we set both of those factors equal to zero and solve for 𝑥. For the first factor, 𝑥 plus one, we need to subtract one from both sides. And we get 𝑥 equals negative one. For the factor 𝑥 minus four, we add four to both sides. And we see that 𝑥 equals positive four. And so, we can say that the solutions of this quadratic equation are 𝑥 equal to negative one and 𝑥 equals four.
