Question

match the items.

a. real and distinct
b. \\(x^2 + 6x - 7 = 0\\)
c. real and equal
d. complex
e. \\(x^2 + 4x + 4 = 0\\)

1. \\(2x^2 - 7x + 6 = 0\\)
2. \\(x^2 - x = -4\\)
3. \\(3x^2 + 6x + 3 = 0\\)
4. real and distinct
5. real and equal

Explanation:

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<reasoning>

Analyze the discriminant of the first equation

Using the discriminant formula \(\Delta = b^2 - 4ac\) for \(2x^2 - 7x + 6 = 0\):
\[
\Delta = (-7)^2 - 4(2)(6) = 49 - 48 = 1 > 0
\]
Since \(\Delta > 0\), the roots are real and distinct.

Analyze the discriminant of the second equation

Rewrite \(x^2 - x = -4\) as \(x^2 - x + 4 = 0\):
\[
\Delta = (-1)^2 - 4(1)(4) = 1 - 16 = -15 < 0
\]
Since \(\Delta < 0\), the roots are complex.

Analyze the discriminant of the third equation

For \(3x^2 + 6x + 3 = 0\):
\[
\Delta = 6^2 - 4(3)(3) = 36 - 36 = 0
\]
Since \(\Delta = 0\), the roots are real and equal.

Match the remaining items

For item 4, "Real and Distinct" matches the equation \(x^2 + 6x - 7 = 0\) (since \(\Delta = 6^2 - 4(1)(-7) = 64 > 0\)), which is option b.
For item 5, "Real and Equal" matches the equation \(x^2 + 4x + 4 = 0\) (since \(\Delta = 4^2 - 4(1)(4) = 0\)), which is option e.
</reasoning>

<answer>

No.	Answer
2	d. Complex
3	c. Real and Equal
4	b. \(x^2 + 6x - 7 = 0\)
5	e. \(x^2 + 4x + 4 = 0\)

</answer>

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Answer:

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<reasoning>

Analyze the discriminant of the first equation

Using the discriminant formula \(\Delta = b^2 - 4ac\) for \(2x^2 - 7x + 6 = 0\):
\[
\Delta = (-7)^2 - 4(2)(6) = 49 - 48 = 1 > 0
\]
Since \(\Delta > 0\), the roots are real and distinct.

Analyze the discriminant of the second equation

Rewrite \(x^2 - x = -4\) as \(x^2 - x + 4 = 0\):
\[
\Delta = (-1)^2 - 4(1)(4) = 1 - 16 = -15 < 0
\]
Since \(\Delta < 0\), the roots are complex.

Analyze the discriminant of the third equation

For \(3x^2 + 6x + 3 = 0\):
\[
\Delta = 6^2 - 4(3)(3) = 36 - 36 = 0
\]
Since \(\Delta = 0\), the roots are real and equal.

Match the remaining items

For item 4, "Real and Distinct" matches the equation \(x^2 + 6x - 7 = 0\) (since \(\Delta = 6^2 - 4(1)(-7) = 64 > 0\)), which is option b.
For item 5, "Real and Equal" matches the equation \(x^2 + 4x + 4 = 0\) (since \(\Delta = 4^2 - 4(1)(4) = 0\)), which is option e.
</reasoning>

<answer>

No.	Answer
2	d. Complex
3	c. Real and Equal
4	b. \(x^2 + 6x - 7 = 0\)
5	e. \(x^2 + 4x + 4 = 0\)

</answer>

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