Answers:
- 5x^2 + 2 = 4x has no real solutions
- 3x^2 + 24x = -48 has exactly one real solution
- 3(x+5)^2 = -2 has no real solutions
- 4x^2 - 16x = 0 has two real solutions
======================================================
Step-by-step explanation:
We can use the discriminant formula to determine how many (and what type of) roots we have. The term "root" is another word for "solution" or "x intercept". It's the x value that makes y = 0.
Any quadratic is of the form ax^2 + bx + c = 0
The discriminant formula is
d = b^2 - 4ac
which is found underneath the square root in the quadratic formula.
Here are the three possible outcomes
- d > 0: we get two different real number solutions
- d = 0: we have exactly one real number solution
- d < 0: there are no real number solutions (the two solutions are complex or imaginary numbers).
With all that in mind, let's find the discriminant of the first equation on the very left.
First we need to get it into standard form ax^2+bx+c = 0
5x^2 + 2 = 4x
5x^2 - 4x + 2 = 0
We see that
Computing the discriminant gets us...
d = b^2 - 4ac
d = (-4)^2 - 4(5)(2)
d = 16 - 40
d = -24
The actual discriminant value itself doesn't matter. All we care about is whether its positive, zero, or negative.
In this case, we get a negative discriminant. So this equation goes to the "no real solutions" column.
-------------------------
Let's move onto the next quadratic
3x^2 + 24x = -48
3x^2 + 24x + 48 = 0
a = 3, b = 24, c = 48
d = b^2 - 4ac
d = 24^2 - 4(3)(48)
d = 0
The discriminant of zero means this equation goes into the "exactly one real solution" category.
-------------------------
For the next equation, we don't need to use the discriminant formula here.
Notice that the right hand side is negative, but the left side is never negative because 3 is positive and (x+5)^2 is never negative. There is no way to have 3(x+5)^2 be negative if we let x be a real number only.
The equation 3(x+5)^2 = -2 goes into "no real solutions" pile.
-------------------------
For the last equation, we have
a = 4, b = -16, c = 0
So,
d = b^2 - 4ac
d = (-16)^2 - 4(4)(0)
d = 256
The result is a positive number, so we have two different solutions.
The discriminant is a perfect square which means we not only have real number solutions, but those solutions are rational numbers. This only works when a,b,c are rational as well.