Questions: Solve the equation using the quadratic formula. x^2+9x+4=0 The solution set is

Transcript text: Solve the equation using the quadratic formula. \[ x^{2}+9 x+4=0 \] The solution set is $\square$

Solution

Solution Steps

To solve the quadratic equation $x^2 + 9x + 4 = 0$ using the quadratic formula, we identify the coefficients $a = 1$, $b = 9$, and $c = 4$. The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

We will substitute the values of $a$, $b$, and $c$ into this formula to find the solutions for $x$.

Step 1: Identify the Coefficients

The given quadratic equation is $x^2 + 9x + 4 = 0$. We identify the coefficients as follows:

$a = 1$
$b = 9$
$c = 4$

Step 2: Calculate the Discriminant

The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by: \[ \Delta = b^2 - 4ac \] Substituting the values, we get: \[ \Delta = 9^2 - 4 \times 1 \times 4 = 81 - 16 = 65 \]

Step 3: Apply the Quadratic Formula

The quadratic formula is: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of $a$, $b$, and $\Delta$, we find: \[ x = \frac{-9 \pm \sqrt{65}}{2} \]

Step 4: Calculate the Roots

Calculate the two possible values for $x$: \[ x_1 = \frac{-9 + \sqrt{65}}{2} \approx -0.4689 \] \[ x_2 = \frac{-9 - \sqrt{65}}{2} \approx -8.5311 \]