simplify the expression $\frac{-b + \sqrt{b^2 - 4ac}}{2a}$ given that $a = 1$, $b = -18$, and $c = -3$.

$\frac{-b + \sqrt{b^2 - 4ac}}{2a} = \square$
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals and $i$ as needed.)

Question

simplify the expression $\frac{-b + \sqrt{b^2 - 4ac}}{2a}$ given that $a = 1$, $b = -18$, and $c = -3$.

$\frac{-b + \sqrt{b^2 - 4ac}}{2a} = \square$
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals and $i$ as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Formula", "Simplifying Radicals" ], "new_concepts": [], "current_concepts": [ "Quadratic Formula", "Simplifying Radicals", "Algebraic Expressions" ] } ## Substitute the given values into the expression $$ \begin{aligned} a &= 1, \quad b = -18, \quad c = -3 \ \frac{-b + \sqrt{b^2 - 4ac}}{2a} &= \frac{-(-18) + \sqrt{(-18)^2 - 4(1)(-3)}}{2(1)} \end{aligned} $$ ## Simplify the terms inside the radical and the denominator $$ \begin{aligned} -(-18) &= 18 \ (-18)^2 &= 324 \ -4(1)(-3) &= 12 \ b^2 - 4ac &= 324 + 12 = 336 \ 2a &= 2 \ \frac{18 + \sqrt{336}}{2} \end{aligned} $$ ## Simplify the radical and divide by the denominator $$ \begin{aligned} \sqrt{336} &= \sqrt{16 \times 21} = 4\sqrt{21} \ \frac{18 + 4\sqrt{21}}{2} &= \frac{18}{2} + \frac{4\sqrt{21}}{2} = 9 + 2\sqrt{21} \end{aligned} $$ Simplify the expression $\frac{-b + \sqrt{b^2 - 4ac}}{2a}$ given that $a = 1$, $b = -18$, and $c = -3$. $\frac{-b + \sqrt{b^2 - 4ac}}{2a} =$ $9 + 2\sqrt{21}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Substitute the given values into the expression

Simplify the terms inside the radical and the denominator

Simplify the radical and divide by the denominator

Answer: