For which values of a do y = x^2 − 2x + a have zero places?
(Sorry if this is translated wrong, it's nollställen in Swedish, which could be zeros.. Quadratic Function maybe?)
I no that for real root
b²≥4ac
If
B²=4ac for equal root
Compare
x²-2x+a=ax²+bx+c
a=1 b=-2 c=a
(2)²=4(a)(a)²
4=4a²
a²=1
a=1
See (1)²-2(1)+1=2-2=0
Correction
2²=4(a)(a)
4=4a²
Not
2²=4(a)(a)²
For real zeros,
b^2 - 4ac ≥ 0
4 - 4(1)(a) ≥ 0
1 - a ≥ 0
-a ≥ -1
a ≤ 1
The confusion arises with the use of "a" , which is used in the definition of
b^2 - 4ac for ax^2 + bx + c = 0
A better wording might have been:
For which values of k do y = x^2 − 2x + k have zero places?
notice in the original we would have
a = 1
b = -2
c = a
Yes, you're correct! "Nollställen" in Swedish translates to "zeros" in English when talking about quadratic functions.
To find the values of a for which the quadratic function y = x^2 - 2x + a has zero places (or no real solutions), we need to consider the discriminant of the quadratic equation.
The discriminant, denoted as Δ, is a formula used to determine the nature of the solutions of a quadratic equation. For the quadratic equation ax^2 + bx + c = 0, the discriminant is given by Δ = b^2 - 4ac.
In this case, the quadratic function is y = x^2 - 2x + a. Comparing it to the standard form (ax^2 + bx + c), we have a = 1, b = -2, and c = a.
To have zero places (no real solutions), the discriminant Δ must be negative.
Thus, we can solve the inequality b^2 - 4ac < 0. Substituting the values, we get (-2)^2 - 4(1)(a) < 0.
Simplifying further, we have 4 - 4a < 0.
Now, solve this inequality by isolating the variable a:
4 - 4a < 0
4 < 4a
1 < a
Therefore, the values of a for which the quadratic function y = x^2 - 2x + a has zero places (no real solutions) are when a is greater than 1. In other words, the set of values of a is [1, +∞).