Question
Use the discriminant to determine the number of real solutions of 3x2+6x−42=0
(1 point)
Responses
2
2
0
0
1
1
3
(1 point)
Responses
2
2
0
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1
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3
Answers
Answered by
GPT-4o mini
To determine the number of real solutions of the quadratic equation \(3x^2 + 6x - 42 = 0\) using the discriminant, we can use the formula for the discriminant \(D\):
\[
D = b^2 - 4ac
\]
where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).
In this case, we have:
- \(a = 3\)
- \(b = 6\)
- \(c = -42\)
Now, we can calculate the discriminant:
\[
D = 6^2 - 4(3)(-42)
\]
\[
D = 36 - 4(3)(-42)
\]
\[
D = 36 - (-504)
\]
\[
D = 36 + 504
\]
\[
D = 540
\]
Since \(D > 0\), this indicates that there are **2 distinct real solutions** to the equation \(3x^2 + 6x - 42 = 0\).
Therefore, the answer is:
**2**
\[
D = b^2 - 4ac
\]
where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).
In this case, we have:
- \(a = 3\)
- \(b = 6\)
- \(c = -42\)
Now, we can calculate the discriminant:
\[
D = 6^2 - 4(3)(-42)
\]
\[
D = 36 - 4(3)(-42)
\]
\[
D = 36 - (-504)
\]
\[
D = 36 + 504
\]
\[
D = 540
\]
Since \(D > 0\), this indicates that there are **2 distinct real solutions** to the equation \(3x^2 + 6x - 42 = 0\).
Therefore, the answer is:
**2**