How many real number solutions does the equation have
y=-4x^2+7x-8
none
one
two
infinite
How many real number solutions does the equation have
y=3x^2+18x+27
none
one
two
infinite
Thank you!
bru why dont mfs just add the whole testπ
You state the equation as a function, so to speak of real number solutions makes no sense.
There are an infinite number of solutions, any ordered pair satisfying the equation would be one.
You probably want the solution to
-4x^2 + 7x - 8 = 0
in both cases, it depends on the value of the discriminant b^2 - 4ac
for the first:
b^2 - 4ac
= 49-4(-4)(-8) = negative
So -4x^2 + 7x - 8 = 0 has no real solution
here is your graph, notice that the curve does not reach the x-axis
http://www.wolframalpha.com/input/?i=plot+y%3D-4x%5E2%2B7x-8+
for 3x^2 + 18x + 27 = 0
b^2- 4ac
= 18^2 - 4(3)(27) = 0
so there is ONE real solution
verification:
http://www.wolframalpha.com/input/?i=plot+y%3D3x%5E2%2B18x%2B27
graph touches the x-axis once, so one solution.
You should make yourself familiar to the properties of solutions of a quadratic based on the sign of b^2 - 4ac
This should be found in your text or your notes.
Oh, equations! They're like puzzles, but with numbers. Alright, let me put on my mathematician's nose and give these questions a shot.
For the first equation, y = -4x^2 + 7x - 8, we need to find out how many real number solutions it has. Well, if you grab your detective hat and some fancy math skills, you'll notice that it's a quadratic equation in the form of y = ax^2 + bx + c. Now, the number of real solutions can be determined by something called the discriminant, which is b^2 - 4ac. If the discriminant is greater than zero, we'll have two real solutions. If it's equal to zero, we'll have one real solution. And if it's less than zero, we'll have none. So, let's do some sleuthing.
For y = -4x^2 + 7x - 8, the discriminant is 7^2 - 4(-4)(-8), which simplifies to 49 - 128, giving us -79. Uh-oh, that's less than zero. It seems we're 79 steps away from finding any real solutions. So, the answer is none.
Now, for the second equation, y = 3x^2 + 18x + 27, let's put on our detective hat once more. The discriminant in this case is 18^2 - 4(3)(27), which gives us 324 - 324. Hmm, that simplifies to zero. Ah-ha! We've found a real solution, my friend. Just one though, don't get too greedy.
So, to sum it up, the first equation has none (0) real solutions, while the second equation has one (1) real solution. You're welcome!
To determine the number of real number solutions for quadratic equations, we can analyze the discriminant of the equation. The discriminant is a term in the quadratic formula which can give us information about the nature of the solutions.
For the equation y = -4x^2 + 7x - 8:
The quadratic equation is in the form of ax^2 + bx + c = 0, where a = -4, b = 7, and c = -8.
The discriminant, denoted as Ξ, is given by Ξ = b^2 - 4ac.
Substituting the values, we have: Ξ = (7)^2 - 4(-4)(-8) = 49 - 128 = -79.
Since the discriminant is negative (Ξ < 0), the equation has no real number solutions. Therefore, the correct answer is none.
For the equation y = 3x^2 + 18x + 27:
Again, the quadratic equation is in the form of ax^2 + bx + c = 0, where a = 3, b = 18, and c = 27.
Calculating the discriminant: Ξ = (18)^2 - 4(3)(27) = 324 - 324 = 0.
As the discriminant is zero (Ξ = 0), it means the equation has exactly one real number solution. Therefore, the correct answer is one.
In summary:
- For the equation y = -4x^2 + 7x - 8, the answer is none.
- For the equation y = 3x^2 + 18x + 27, the answer is one.