Solve using the quadratic formula. $\newline$ $n^2 - 8n + 3 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $n =$ _ or $n =$ _

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Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

n^2 - 8n + 3 = 0

\newline

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

\newline

n =

_____ or

n =

_____

Identify coefficients: Identify coefficients $a$ , $b$ , and $c$ from the quadratic equation $an^2 + bn + c = 0$ . $\newline$ Here, $a = 1$ , $b = -8$ , and $c = 3$ .
Write quadratic formula: Write down the quadratic formula: $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Plug values into formula: Plug the values of $a$ , $b$ , and $c$ into the quadratic formula. $n = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$ .
Simplify inside square root: Simplify inside the square root: $(-8)^2 - 4\cdot1\cdot3 = 64 - 12$ . $\newline$ $n = \frac{8 \pm \sqrt{64 - 12}}{2}$ .
Further simplify square root: Further simplify under the square root: $64 - 12 = 52$ . $\newline$ $n = (8 \pm \sqrt{52}) / 2$ .
Simplify square root of $52$ : Simplify the square root of $52$ . $\sqrt{52}$ is not a perfect square, so it remains as $\sqrt{52}$ . $n = \frac{8 \pm \sqrt{52}}{2}$ .
Divide by $2$ : Divide both terms in the numerator by $2$ . $\newline$ $n = \frac{4 \pm \sqrt{52}}{2}$ .
Calculate two possible values: Simplify $\sqrt{52} / 2$ . $\newline$ Since $\sqrt{52}$ is not a perfect square, we can't simplify this further without a calculator. $\newline$ $n = 4 \pm \sqrt{52} / 2$ .
Approximate square root: Calculate the two possible values for $n$ . $\newline$ First value: $n = \frac{4 + \sqrt{52}}{2}$ . $\newline$ Second value: $n = \frac{4 - \sqrt{52}}{2}$ .
Find approximate solutions: Use a calculator to approximate $\sqrt{52} / 2$ . $\newline$ $\sqrt{52} / 2 \approx 3.61$ . $\newline$ $n = 4 \pm 3.61$ .
Find approximate solutions: Use a calculator to approximate $\sqrt{52} / 2$ . $\newline$ $\sqrt{52} / 2 \approx 3.61$ . $\newline$ $n = 4 \pm 3.61$ .Find the two approximate solutions for $n$ . $\newline$ First value: $n \approx 4 + 3.61 = 7.61$ . $\newline$ Second value: $n \approx 4 - 3.61 = 0.39$ .