Determine the type of the roots of the equation four 𝑥 times 𝑥 plus five equals negative 25.
In order to determine these type of roots, we can look at the discriminant, which is 𝑏 squared minus four 𝑎𝑐. If it ends up being less than zero, it will have two different complex and nonreal roots. If 𝑏 squared minus four 𝑎𝑐 ends up being equal to zero, it will have two real and equal roots. And if it’s greater than zero, it will have two different real roots.
This comes from the quadratic formula, 𝑏 squared minus four 𝑎𝑐. So if you think about — you know if you get a number underneath the square root and it’s less than zero, that means it’s negative. So you’re gonna have complex imaginary numbers that you’re gonna be working with.
If it would be equal to zero, the square root would completely disappear and you’re gonna have an answer that’s just one answer because the square has disappeared. So it’s just gonna be whatever the fraction is. And then if it’s greater than zero, it’s gonna be some positive number and it may be a perfect square or it may be a nonperfect square. It just depends.
So that’s where we get these from. So the first thing that we need to do is to distribute four 𝑥. And now we should add 25 to both sides. So we have four 𝑥 squared plus 20𝑥 plus 25 equals zero. Therefore, 𝑎 is four, 𝑏 is 20, and 𝑐 is 25.
Therefore, 𝑏 squared minus four 𝑎𝑐 will be equal to 20 squared minus four times four times 25, which is 400 minus 400, which is equal to zero. So we will have real and equal roots.