Certainly! Let's solve the problem step by step.
First, examine the given quadratic formula expression:
[tex]\[
\frac{1 \pm \sqrt{(-1)^2-4(3)(4)}}{2(3)}
\][/tex]
### Step 1: Identify the discriminant
The discriminant in a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\(b^2 - 4ac\)[/tex].
From the expression:
[tex]\[
(-1)^2 - 4(3)(4)
\][/tex]
### Step 2: Calculate the discriminant
[tex]\[
(-1)^2 = 1
\][/tex]
[tex]\[
4 \cdot 3 \cdot 4 = 48
\][/tex]
[tex]\[
1 - 48 = -47
\][/tex]
The discriminant is [tex]\(-47\)[/tex].
### Step 3: Match the discriminant with the quadratic equations
Now, we need to find which of the given quadratic equations results in a discriminant of [tex]\(-47\)[/tex].
Given equations:
1. [tex]\(x^2 + 4 = -3x \rightarrow x^2 + 3x + 4 = 0 \rightarrow a = 1, b = 3, c = 4\)[/tex]
2. [tex]\(3x^2 + 4 = -x \rightarrow 3x^2 + x + 4 = 0 \rightarrow a = 3, b = 1, c = 4\)[/tex]
3. [tex]\(4x^2 + 3 = x \rightarrow 4x^2 - x + 3 = 0 \rightarrow a = 4, b = -1, c = 3\)[/tex]
4. [tex]\(3x^2 + 4 = x \rightarrow 3x^2 - x + 4 = 0 \rightarrow a = 3, b = -1, c = 4\)[/tex]
5. [tex]\(x^2 + 4 = 3x \rightarrow x^2 - 3x + 4 = 0 \rightarrow a = 1, b = -3, c = 4\)[/tex]
We compute the discriminants for each given quadratic equation:
1. For [tex]\(x^2 + 3x + 4 = 0\)[/tex]:
[tex]\[
b^2 - 4ac = 3^2 - 4(1)(4) = 9 - 16 = -7
\][/tex]
2. For [tex]\(3x^2 + x + 4 = 0\)[/tex]:
[tex]\[
b^2 - 4ac = 1^2 - 4(3)(4) = 1 - 48 = -47
\][/tex]
3. For [tex]\(4x^2 - x + 3 = 0\)[/tex]:
[tex]\[
b^2 - 4ac = (-1)^2 - 4(4)(3) = 1 - 48 = -47
\][/tex]
4. For [tex]\(3x^2 - x + 4 = 0\)[/tex]:
[tex]\[
b^2 - 4ac = (-1)^2 - 4(3)(4) = 1 - 48 = -47
\][/tex]
5. For [tex]\(x^2 - 3x + 4 = 0\)[/tex]:
[tex]\[
b^2 - 4ac = (-3)^2 - 4(1)(4) = 9 - 16 = -7
\][/tex]
### Match the Results
We see that the discriminant [tex]\(-47\)[/tex] matches the following equations:
- [tex]\(3x^2 + x + 4 = 0\)[/tex]
- [tex]\(4x^2 - x + 3 = 0\)[/tex]
- [tex]\(3x^2 - x + 4 = 0\)[/tex]
Corresponding to these equations:
- [tex]\(3x^2 + 4 = -x\)[/tex]
- [tex]\(4x^2 + 3 = x\)[/tex]
- [tex]\(3x^2 + 4 = x\)[/tex]
### Final Answer
Therefore, the quadratic equations whose solutions are given by the expression:
[tex]\[
\frac{1 \pm \sqrt{(-1)^2-4(3)(4)}}{2(3)}
\][/tex]
are:
[tex]\[
3 x^2 + 4 = -x
\][/tex]
and
[tex]\[
3 x^2 + 4 = x
\][/tex]
