Solve using the quadratic formula. $\newline$ $4k^2 - 9k + 4 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $k =$ _ or $k =$ _

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Algebra 2

Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

4k^2 - 9k + 4 = 0

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

\newline

k =

_____ or

k =

_____

Identify coefficients: Identify the coefficients of the quadratic equation. $\newline$ The quadratic equation is in the form $ax^2 + bx + c = 0$ . For the equation $4k^2 - 9k + 4 = 0$ , the coefficients are: $\newline$ a = $4$ $\newline$ b = $-9$ $\newline$ c = $4$
Write formula: Write down the quadratic formula. $\newline$ The quadratic formula is given by: $\newline$ $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Substitute values: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula. $\newline$ Substitute $a = 4$ , $b = -9$ , and $c = 4$ into the quadratic formula: $\newline$ $k = \frac{-(-9) \pm \sqrt{(-9)^2 - 4\cdot4\cdot4}}{2\cdot4}$ $\newline$ $k = \frac{9 \pm \sqrt{81 - 64}}{8}$
Simplify and solve: Simplify under the square root and solve for $k$ . $\newline$ Calculate the value under the square root: $\newline$ $\sqrt{81 - 64} = \sqrt{17}$ $\newline$ Now, substitute this back into the formula: $\newline$ $k = (9 \pm \sqrt{17}) / 8$
Calculate solutions: Calculate the two possible solutions for $k$ . $\newline$ First solution: $\newline$ $k = \frac{9 + \sqrt{17}}{8}$ $\newline$ Second solution: $\newline$ $k = \frac{9 - \sqrt{17}}{8}$
Round and simplify: Simplify the solutions and, if necessary, round to the nearest hundredth. $\newline$ First solution: $\newline$ $k = \frac{9 + \sqrt{17}}{8} \approx \frac{9 + 4.12}{8} \approx \frac{13.12}{8} \approx 1.64$ $\newline$ Second solution: $\newline$ $k = \frac{9 - \sqrt{17}}{8} \approx \frac{9 - 4.12}{8} \approx \frac{4.88}{8} \approx 0.61$