Solve the equation using the quadratic formula. $$ x^{2}+11 \mathrm{x}+4=0 $$ The solution set is (Simplify your answer. Type an exact answer, using radicals and $$ i $$ as needed. Use in answers as needed.)

Question

UpStudy AI · Accepted Answer

We are given the quadratic equation

$$
x^{2} + 11x + 4 = 0.
$$

### Step 1: Identify the coefficients

The quadratic equation is in the form

$$
ax^2 + bx + c = 0,
$$

so we have:

- $$a = 1$$
- $$b = 11$$
- $$c = 4$$

### Step 2: Write down the quadratic formula

The quadratic formula is

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$

### Step 3: Substitute the coefficients into the quadratic formula

Substitute $$a = 1$$, $$b = 11$$, and $$c = 4$$ into the formula:

$$
x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}.
$$

### Step 4: Simplify the expression inside the square root

Calculate the discriminant:

$$
11^2 - 4 \cdot 1 \cdot 4 = 121 - 16 = 105.
$$

Now the expression becomes:

$$
x = \frac{-11 \pm \sqrt{105}}{2}.
$$

### Step 5: Write the solution set

The solutions to the equation are

$$
x = \frac{-11 + \sqrt{105}}{2} \quad \text{and} \quad x = \frac{-11 - \sqrt{105}}{2}.
$$

Thus, the solution set is

$$
\left\{ \frac{-11 + \sqrt{105}}{2}, \frac{-11 - \sqrt{105}}{2} \right\}.
$$
The solutions are $$ x = \frac{-11 + \sqrt{105}}{2} $$ and $$ x = \frac{-11 - \sqrt{105}}{2} $$.

Solve the equation using the quadratic formula. \[ x^{2}+11 \mathrm{x}+4=0 \] The solution set is (Simplify your answer. Type an exact answer, using radicals and \( i \) as needed. Use in answers as needed.)

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