QUESTION IMAGE
Question
for the following quadratic equation, find the discriminant.
\\5x^2 - 22x + 87 = -6x - 8\\
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<reasoning>
Write the equation in standard form
To find the discriminant, we first rewrite the given quadratic equation in standard form:
\[ax^2 + bx + c = 0\]
We start with the given equation:
\[5x^2 - 22x + 87 = -6x - 8\]
Add \(6x\) and \(8\) to both sides of the equation:
\[5x^2 - 22x + 6x + 87 + 8 = 0\]
\[5x^2 - 16x + 95 = 0\]
Identify the coefficients
From the standard form \(5x^2 - 16x + 95 = 0\), we identify the coefficients:
\[a = 5\]
\[b = -16\]
\[c = 95\]
Calculate the discriminant
The formula for the discriminant \(D\) of a quadratic equation is:
\[D = b^2 - 4ac\]
Substitute the identified coefficients into the formula:
\[D = (-16)^2 - 4(5)(95)\]
\[D = 256 - 1900\]
\[D = -1644\]
</reasoning>
<answer>
\(-1644\)
</answer>
<post_analysis>
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"Mathematics",
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</pre_analysis>
<reasoning>
Write the equation in standard form
To find the discriminant, we first rewrite the given quadratic equation in standard form:
\[ax^2 + bx + c = 0\]
We start with the given equation:
\[5x^2 - 22x + 87 = -6x - 8\]
Add \(6x\) and \(8\) to both sides of the equation:
\[5x^2 - 22x + 6x + 87 + 8 = 0\]
\[5x^2 - 16x + 95 = 0\]
Identify the coefficients
From the standard form \(5x^2 - 16x + 95 = 0\), we identify the coefficients:
\[a = 5\]
\[b = -16\]
\[c = 95\]
Calculate the discriminant
The formula for the discriminant \(D\) of a quadratic equation is:
\[D = b^2 - 4ac\]
Substitute the identified coefficients into the formula:
\[D = (-16)^2 - 4(5)(95)\]
\[D = 256 - 1900\]
\[D = -1644\]
</reasoning>
<answer>
\(-1644\)
</answer>
<post_analysis>
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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Discriminant"
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