Question
The equation y=x2−4x+5
𝑦
=
𝑥
2
−
4
𝑥
+
5
has two solutions. Compare this information with the graph of the equation. How many real and non-real solutions does the equation have?
(1 point)
Responses
two real and zero non-real
two real and zero non-real
zero real and zero non-real
zero real and zero non-real
one real and one non-real
one real and one non-real
zero real and two non-real
𝑦
=
𝑥
2
−
4
𝑥
+
5
has two solutions. Compare this information with the graph of the equation. How many real and non-real solutions does the equation have?
(1 point)
Responses
two real and zero non-real
two real and zero non-real
zero real and zero non-real
zero real and zero non-real
one real and one non-real
one real and one non-real
zero real and two non-real
Answers
GPT-4o mini
To determine the number of real and non-real solutions for the equation \( y = x^2 - 4x + 5 \), we can analyze it using the discriminant from the quadratic formula. The standard form of a quadratic equation is \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -4 \)
- \( c = 5 \)
The discriminant \( D \) is calculated using the formula:
\[
D = b^2 - 4ac
\]
Substituting in our values:
\[
D = (-4)^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4
\]
Since the discriminant is negative (\( D < 0 \)), this indicates that there are no real solutions and, therefore, the equation has two non-real solutions.
Thus, the correct response is:
**zero real and two non-real**.
- \( a = 1 \)
- \( b = -4 \)
- \( c = 5 \)
The discriminant \( D \) is calculated using the formula:
\[
D = b^2 - 4ac
\]
Substituting in our values:
\[
D = (-4)^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4
\]
Since the discriminant is negative (\( D < 0 \)), this indicates that there are no real solutions and, therefore, the equation has two non-real solutions.
Thus, the correct response is:
**zero real and two non-real**.