Solve using the quadratic formula. $\newline$ $4k^2 - k - 8 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $k =$ _ or $k =$ _

Full solution

Q. Solve using the quadratic formula.

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4k^2 - k - 8 = 0

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

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k =

_____ or

k =

_____

Identify coefficients: Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $4k^2 − k − 8 = 0$ by comparing it to the standard form $ax^2 + bx + c = 0$ . $\newline$ $a = 4$ , $b = -1$ , $c = -8$
Substitute values into formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula, $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . $k = \frac{-(-1) \pm \sqrt{(-1)^2 - 4\cdot4\cdot(-8)}}{2\cdot4}$
Simplify expression and constants: Simplify the expression inside the square root and the constants outside the square root. $\newline$ $k = \frac{1 \pm \sqrt{1 + 128}}{8}$ $\newline$ $k = \frac{1 \pm \sqrt{129}}{8}$
Calculate square root of $129$ : Calculate the square root of $129$ to find the two possible solutions for $k$ . $\sqrt{129} \approx 11.36$ (rounded to two decimal places) $k = \frac{1 + 11.36}{8}$ or $k = \frac{1 - 11.36}{8}$
Solve for two values of $k$ : Solve for the two values of $k$ . $k \approx (1 + 11.36) / 8$ or $k \approx (1 - 11.36) / 8$ $k \approx 12.36 / 8$ or $k \approx -10.36 / 8$ $k \approx 1.55$ or $k \approx -1.295$