## Precalculus (6th Edition)

Francesca was wrong because the quadratic formula works even when $b=0$.
RECALL: The quadratic equation $ax^2+bx+c=0$ can be solved using the quadratic formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ The given equation has: $a=1, b=0, c=-19$ Substitute these values into the quadratic formula to obtain: $x=\dfrac{-0 \pm \sqrt{0^2-4(1)(-19)}}{2(1)} \\x=\dfrac{\pm\sqrt{0+76}}{2} \\x=\dfrac{\pm\sqrt{76}}{2} \\x=\dfrac{\pm\sqrt{4(19)}}{2} \\x=\dfrac{2\sqrt{19}}{2} \\x=\pm\sqrt{19}$ Thus, the solution set is $\color{blue}{\left\{-\sqrt{19}, \sqrt{19}\right\}}$. Francesca was wrong because the quadratic formula actually works regardless of whether $b=0$ or not.
The passage explains why a quadratic formula can be used even when \( b = 0 \) and provides a detailed solution to a specific quadratic equation.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
## Precalculus (6th Edition)

Francesca was wrong because the quadratic formula works even when $b=0$.
RECALL: The quadratic equation $ax^2+bx+c=0$ can be solved using the quadratic formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ The given equation has: $a=1, b=0, c=-19$ Substitute these values into the quadratic formula to obtain: $x=\dfrac{-0 \pm \sqrt{0^2-4(1)(-19)}}{2(1)} \\x=\dfrac{\pm\sqrt{0+76}}{2} \\x=\dfrac{\pm\sqrt{76}}{2} \\x=\dfrac{\pm\sqrt{4(19)}}{2} \\x=\dfrac{2\sqrt{19}}{2} \\x=\pm\sqrt{19}$ Thus, the solution set is $\color{blue}{\left\{-\sqrt{19}, \sqrt{19}\right\}}$. Francesca was wrong because the quadratic formula actually works regardless of whether $b=0$ or not.
user
The passage explains why a quadratic formula can be used even when \( b = 0 \) and provides a detailed solution to a specific quadratic equation.

TYPE: SINGLE_QUESTION_WITH_ANSWER