Video: Determining the Types of Roots of Quadratic Equations

Determine the type of the roots of the equation 4๐‘ฅ(๐‘ฅ + 5) = โˆ’25.

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Video Transcript

Determine the type of the roots of the equation four ๐‘ฅ times ๐‘ฅ plus five equals negative 25.

In order to determine these type of roots, we can look at the discriminant, which is ๐‘ squared minus four ๐‘Ž๐‘. If it ends up being less than zero, it will have two different complex and nonreal roots. If ๐‘ squared minus four ๐‘Ž๐‘ ends up being equal to zero, it will have two real and equal roots. And if itโ€™s greater than zero, it will have two different real roots.

This comes from the quadratic formula, ๐‘ squared minus four ๐‘Ž๐‘. So if you think about โ€” you know if you get a number underneath the square root and itโ€™s less than zero, that means itโ€™s negative. So youโ€™re gonna have complex imaginary numbers that youโ€™re gonna be working with.

If it would be equal to zero, the square root would completely disappear and youโ€™re gonna have an answer thatโ€™s just one answer because the square has disappeared. So itโ€™s just gonna be whatever the fraction is. And then if itโ€™s greater than zero, itโ€™s gonna be some positive number and it may be a perfect square or it may be a nonperfect square. It just depends.

So thatโ€™s where we get these from. So the first thing that we need to do is to distribute four ๐‘ฅ. And now we should add 25 to both sides. So we have four ๐‘ฅ squared plus 20๐‘ฅ plus 25 equals zero. Therefore, ๐‘Ž is four, ๐‘ is 20, and ๐‘ is 25.

Therefore, ๐‘ squared minus four ๐‘Ž๐‘ 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.

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