Questions: Use the quadratic formula to solve for (x). [4 x^2+9 x+3=0] (If there is more than one solution, separate them with commas.) [x=]

Transcript text: Use the quadratic formula to solve for $x$. \[ 4 x^{2}+9 x+3=0 \] (If there is more than one solution, separate them with commas.) \[ x= \]

Solution

Solution Steps

To solve the quadratic equation $4x^2 + 9x + 3 = 0$ using the quadratic formula, we need to identify the coefficients $a$, $b$, and $c$ from the equation $ax^2 + bx + c = 0$. Then, we apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions for $x$.

Step 1: Identify the coefficients

Given the quadratic equation $4x^2 + 9x + 3 = 0$, we identify the coefficients:

$a = 4$
$b = 9$
$c = 3$

Step 2: Calculate the discriminant

The discriminant $\Delta$ is calculated using the formula: \[ \Delta = b^2 - 4ac \] Substituting the values of $a$, $b$, and $c$: \[ \Delta = 9^2 - 4 \cdot 4 \cdot 3 = 81 - 48 = 33 \]

Step 3: Apply the quadratic formula

The solutions for $x$ are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of $a$, $b$, and $\Delta$: \[ x = \frac{-9 \pm \sqrt{33}}{2 \cdot 4} = \frac{-9 \pm \sqrt{33}}{8} \]

Step 4: Calculate the solutions

We calculate the two solutions: \[ x_1 = \frac{-9 + \sqrt{33}}{8} \approx -0.4069 \] \[ x_2 = \frac{-9 - \sqrt{33}}{8} \approx -1.843 \]