Solve using the quadratic formula. $\newline$ $4r^2 + 9r + 3 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $r =$ _ or $r =$ _

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Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

4r^2 + 9r + 3 = 0

\newline

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

\newline

r =

_____ or

r =

_____

Quadratic Formula: The quadratic formula is given by $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the terms in the quadratic equation $ar^2 + br + c = 0$ . In this case, $a = 4$ , $b = 9$ , and $c = 3$ .
Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . Here, it is $9^2 - 4(4)(3)$ .
Find Solutions: Perform the calculation: $81 - 48 = 33$ . The discriminant is $33$ .
Simplify Equation: Now, plug the values of $a$ , $b$ , and the discriminant into the quadratic formula to find the two possible values for $r$ . $r = \frac{-9 \pm \sqrt{33}}{2 \times 4}$
Irrational Number Solutions: Simplify the equation by performing the operations: $r = \frac{-9 \pm \sqrt{33}}{8}$
Two Solutions: Since the discriminant is positive and not a perfect square, the solutions will be irrational numbers. We can leave the square root as is or approximate the values to the nearest hundredth.
Approximate Square Root: The two solutions are: $r = \frac{-9 + \sqrt{33}}{8}$ and $r = \frac{-9 - \sqrt{33}}{8}$
Calculate Approximate Values: If we want to round to the nearest hundredth, we approximate the square root of $33$ and perform the division: $\newline$ $\sqrt{33} \approx 5.74$ (rounded to the nearest hundredth) $\newline$ $r \approx (-9 + 5.74) / 8$ and $r \approx (-9 - 5.74) / 8$
Calculate Approximate Values: If we want to round to the nearest hundredth, we approximate the square root of $33$ and perform the division: $\newline$ $\sqrt{33} \approx 5.74$ (rounded to the nearest hundredth) $\newline$ $r \approx (-9 + 5.74) / 8$ and $r \approx (-9 - 5.74) / 8$ Calculate the approximate values: $\newline$ $r \approx (-3.26) / 8$ and $r \approx (-14.74) / 8$ $\newline$ $r \approx -0.41$ and $r \approx -1.84$ (rounded to the nearest hundredth)