Determine the type of the roots of the equation two 𝑥 minus four all squared plus 17 equals zero.
So to help us solve this problem, what I’ve first done is rewritten it. So we’ve got two 𝑥 minus four multiplied by two 𝑥 minus four, cause we had two 𝑥 minus four squared plus 17 equals zero. So we can actually see here that what we need to do first is actually expand the parentheses.
As we’re gonna first of all expand the parentheses, what we do first is we’re gonna multiply the first term in each of our parentheses. So that is two 𝑥 multiplied by two 𝑥, which gives us four 𝑥 squared. Then I’m gonna multiply the first term in our first parentheses by the second term in our second parentheses. And that’s gonna be two multiplied by negative four, which gives us negative eight 𝑥.
Remember to take care with the negatives and positives. And then we have another negative eight 𝑥 because we have the second term in the first parentheses multiplied by the first term in the second parentheses. So we have negative four multiplied by two 𝑥, which gives us negative eight 𝑥. And then finally, we multiply the last two terms. So we have negative four multiplied by negative four, which gives us positive 16.
Brilliant! And then finally we’ve still got the plus 17 on the end. And then that’s all equal to zero. Okay, great! So now let’s simplify. So if I simplify, I get four 𝑥 squared. And then I’ve got negative eight 𝑥 minus eight 𝑥, which gives me negative 16𝑥. So we now got four 𝑥 squared minus 16𝑥. And now we have 16 plus 17, so we get plus 33. And this is all equal to zero.
Okay, great! So we’ve now expanded and we’ve simplified. But why have we done this? Why do we want it in this form? Well, we put it in this form because we actually we want these. We want 𝑎, we want 𝑏, and we want 𝑐. We want these because actually we want to use the discriminant. And the discriminant is 𝑏 squared minus four 𝑎𝑐. And why do we want this? Well, we want this because the discriminant is very useful because actually enables us to tell what type of roots an equation has.
So first of all, if we have 𝑏 squared minus four 𝑎𝑐 is greater than zero so therefore is positive, that tells us that our equation is gonna have real and different roots. However, if we have 𝑏 squared minus four 𝑎𝑐 being equal to zero, then this tells us that our equation has real and repeated roots. Or the roots are real and the same.
So we have 𝑏 squared minus four 𝑎𝑐 is less than zero. And if this is the case, then we know that our roots are complex and not real. Okay, great! So we know what the discriminant is. We now know how to use it and what it means. Let’s try and use it to determine the type of the roots of our equation. So therefore we’re gonna use the discriminant.
We’ve got 𝑏 squared minus four 𝑎𝑐. What we do now is we’re gonna substitute our values for 𝑎, 𝑏, and 𝑐 in. So we’re gonna get negative 16 all squared minus four multiplied by four multiplied by 33, which is gonna be equal to 256 minus 528, which gives us a final answer for our discriminant of negative 272.
Okay, great! So we now we got this value. We can actually look back at our description for the discriminant to see what this actually means. Well, if we have a look across, we can actually see that it’s gonna be the third situation, actually gonna be the one that we’re using, because 𝑏 squared minus four 𝑎𝑐 is actually less than zero because we’ve got a negative answer of negative 272.
So therefore we can say that the roots of our equation are complex and not real. And this is because 𝑏 squared minus four 𝑎𝑐 is equal to negative 272, which is less than zero. Okay, great! So we’ve got to our final answer.
Just do a quick recap of what we’ve done. So first of all, make sure that you get your equation into the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, which we did here. And then once you’ve got that, substitute your values for 𝑎, 𝑏, and 𝑐 into the discriminant, which is 𝑏 squared minus four 𝑎𝑐, to get your value, which we did. And we got negative 272.
And then finally, once you’ve got your value for your discriminant, you have to decide, is it greater than, less than, or equal to zero? And this will tell you what roots your equation has.