assuming $b \
eq 0$, does the sign of $b$ affect the value of the discriminant?

no, only when b is negative
yes, only when b is positive
yes
no

question 5 (3 points)
the function $f(x) = x^2 + kx + k + 8$ touches the $x$-axis once. what value(s) could $k$ be?
choose all that apply, if more than one.

$k = \frac{1}{2}$
$k = 8$
$k

Question

assuming $b \
eq 0$, does the sign of $b$ affect the value of the discriminant?

no, only when b is negative
yes, only when b is positive
yes
no

question 5 (3 points)
the function $f(x) = x^2 + kx + k + 8$ touches the $x$-axis once. what value(s) could $k$ be?
choose all that apply, if more than one.

$k = \frac{1}{2}$
$k = 8$
$k < -4$
$k = -4$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Discriminant" ], "new_concepts": [], "current_concepts": [ "Discriminant", "Quadratic Zeroes" ] } ## Analyze the effect of the sign of $b$ on the discriminant $$ \begin{aligned} &D = b^2 - 4ac \ &(-b)^2 = b^2 \ &\text{Therefore, } D \text{ remains unchanged regardless of the sign of } b. \end{aligned} $$ ## Solve for $k$ when the quadratic function touches the x-axis once $$ \begin{aligned} &f(x) = x^2 + kx + (k + 8) \ &a = 1, \quad b = k, \quad c = k + 8 \ &D = b^2 - 4ac = k^2 - 4(1)(k + 8) = k^2 - 4k - 32 \ &D = 0 \implies k^2 - 4k - 32 = 0 \end{aligned} $$ ## Find the roots of the quadratic equation in $k$ $$ \begin{aligned} &(k - 8)(k + 4) = 0 \ &k = 8 \quad \text{or} \quad k = -4 \end{aligned} $$ ## Question 1 no, only when b is negative

yes, only when b is positive

yes

no ## Question 2 $k = \frac{1}{2}$

$k = 8$

$k < -4$

$k = -4$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Discriminant" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the effect of the sign of \(b\) on the discriminant

Solve for \(k\) when the quadratic function touches the x-axis once

Find the roots of the quadratic equation in \(k\)

Answer:

Question 1

Question 2