Question
Use the discriminant to determine the number of real solutions of 5=2x−x
2
.
A. 0 real solutions B. 1 real solution C. 2 real solutions
2
.
A. 0 real solutions B. 1 real solution C. 2 real solutions
Answers
Answered by
GPT-4o mini
To use the discriminant to determine the number of real solutions for the equation \(5 = 2x - x^2\), we first need to rewrite this equation in standard quadratic form.
Starting with:
\[ 5 = 2x - x^2 \]
we can rearrange it into the standard form \(ax^2 + bx + c = 0\):
\[ -x^2 + 2x - 5 = 0 \]
Multiplying through by -1 to make the coefficient of \(x^2\) positive gives:
\[ x^2 - 2x + 5 = 0 \]
Now in standard form, we have:
- \(a = 1\)
- \(b = -2\)
- \(c = 5\)
Next, we calculate the discriminant \(D\) using the formula:
\[
D = b^2 - 4ac
\]
Substituting in the values for \(a\), \(b\), and \(c\):
\[
D = (-2)^2 - 4(1)(5) = 4 - 20 = -16
\]
The discriminant \(D = -16\) is less than 0, which indicates that there are no real solutions.
Thus, the answer is:
**A. 0 real solutions**.
Starting with:
\[ 5 = 2x - x^2 \]
we can rearrange it into the standard form \(ax^2 + bx + c = 0\):
\[ -x^2 + 2x - 5 = 0 \]
Multiplying through by -1 to make the coefficient of \(x^2\) positive gives:
\[ x^2 - 2x + 5 = 0 \]
Now in standard form, we have:
- \(a = 1\)
- \(b = -2\)
- \(c = 5\)
Next, we calculate the discriminant \(D\) using the formula:
\[
D = b^2 - 4ac
\]
Substituting in the values for \(a\), \(b\), and \(c\):
\[
D = (-2)^2 - 4(1)(5) = 4 - 20 = -16
\]
The discriminant \(D = -16\) is less than 0, which indicates that there are no real solutions.
Thus, the answer is:
**A. 0 real solutions**.