Question
Completing the Square & the Quadratic Formula Quick Check
5 of 55 of 5 Items
Question
How many real solutions does the following quadratic equation have?
4x2+x+3=0
(1 point)
Responses
two real solutions
two real solutions
no real solutions
no real solutions
three real solutions
three real solutions
one real solution
one real solution
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Answers
Answered by
GPT-4o mini
To determine how many real solutions the quadratic equation \(4x^2 + x + 3 = 0\) has, we can use the discriminant from the quadratic formula:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\).
For the equation \(4x^2 + x + 3 = 0\):
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now, we calculate the discriminant \(D = b^2 - 4ac\):
\[
D = (1)^2 - 4(4)(3) = 1 - 48 = -47
\]
Since the discriminant \(D\) is negative (\(-47 < 0\)), this means the quadratic equation has **no real solutions**.
Therefore, the correct answer is:
- no real solutions
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\).
For the equation \(4x^2 + x + 3 = 0\):
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now, we calculate the discriminant \(D = b^2 - 4ac\):
\[
D = (1)^2 - 4(4)(3) = 1 - 48 = -47
\]
Since the discriminant \(D\) is negative (\(-47 < 0\)), this means the quadratic equation has **no real solutions**.
Therefore, the correct answer is:
- no real solutions