How many solutions does this have:
x^2 + 2x + 1 = 0
I put one.
9x^2 + 49 = 42x
I put one.
x^2 + 4x + 1 = 0
I put two.
I actually think the 2nd one has no real solutions...
nah, you are correct on all three
9x^2 - 42x + 49 = 0
(3x-7)^2 = 0
Ok cool! you think you can show me how to do these?
1. What is the largest output value of f(x) possible for the following quadratic function? f(x) = - x 2 -6 x + 15.
2. What is the absolute value of the smallest output value of f(x) possible for the following quadratic function? f(x) = x 2 – 2 x - 8
as you know, the vertex of the parabola
y = ax^2 + bx + c
is at x = -b/2a
#1 the parabola opens down, so the vertex is a maximum
#2 has a minimum at the vertex
so, plug in that value for x and let 'er rip
I have no idea what any of that means... :)
To determine the number of solutions for a quadratic equation, we can use the discriminant, which is the expression inside the square root in the quadratic formula. The discriminant is calculated as follows:
Discriminant (D) = b^2 - 4ac
Where "a," "b," and "c" are the coefficients of the quadratic equation ax^2 + bx + c = 0.
Now let's analyze each of the given equations:
1. x^2 + 2x + 1 = 0:
In this equation, a = 1, b = 2, and c = 1.
Calculating the discriminant, we have:
D = (2)^2 - 4(1)(1) = 4 - 4 = 0
Since the discriminant is equal to 0, there is only one solution.
2. 9x^2 + 49 = 42x:
Rearranging the equation to the standard quadratic form (ax^2 + bx + c = 0), we have:
9x^2 - 42x + 49 = 0
Here, a = 9, b = -42, and c = 49.
Calculating the discriminant:
D = (-42)^2 - 4(9)(49) = 1764 - 1764 = 0
Similar to the previous equation, the discriminant is equal to 0, indicating only one solution.
3. x^2 + 4x + 1 = 0:
Here, a = 1, b = 4, and c = 1.
Calculating the discriminant again:
D = (4)^2 - 4(1)(1) = 16 - 4 = 12
Since the discriminant is positive (D > 0), there are two distinct solutions.
Therefore, the number of solutions for each equation is as follows:
- x^2 + 2x + 1 = 0: one solution
- 9x^2 + 49 = 42x: one solution
- x^2 + 4x + 1 = 0: two solutions.