To solve the quadratic equation [tex]\(3 \sqrt{2} x^2 + 5 x + \sqrt{2} = 0\)[/tex], we will follow these steps:
### Step 1: Identify the coefficients
The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. By comparing this with our equation, we find:
- [tex]\(a = 3 \sqrt{2}\)[/tex]
- [tex]\(b = 5\)[/tex]
- [tex]\(c = \sqrt{2}\)[/tex]
### Step 2: Compute the discriminant
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in our values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 5^2 - 4 \cdot 3 \sqrt{2} \cdot \sqrt{2} \][/tex]
[tex]\[ \Delta = 25 - 4 \cdot 3 \cdot 2 \][/tex]
[tex]\[ \Delta = 25 - 24 \][/tex]
[tex]\[ \Delta = 1 \][/tex]
### Step 3: Check the nature of the roots
Since the discriminant ([tex]\(\Delta\)[/tex]) is positive and non-zero, we will have two distinct real roots.
### Step 4: Use the quadratic formula to find the roots
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the formula:
[tex]\[ x = \frac{-5 \pm \sqrt{1}}{2 \cdot 3 \sqrt{2}} \][/tex]
[tex]\[ x = \frac{-5 \pm 1}{2 \cdot 3 \sqrt{2}} \][/tex]
[tex]\[ x = \frac{-5 \pm 1}{6 \sqrt{2}} \][/tex]
### Step 5: Calculate the two solutions
- For the positive square root:
[tex]\[ x_1 = \frac{-5 + 1}{6 \sqrt{2}} \][/tex]
[tex]\[ x_1 = \frac{-4}{6 \sqrt{2}} \][/tex]
[tex]\[ x_1 = \frac{-2}{3 \sqrt{2}} \][/tex]
Rationalizing the denominator, we get:
[tex]\[ x_1 = -\frac{2 \sqrt{2}}{6} \][/tex]
[tex]\[ x_1 = -\frac{\sqrt{2}}{3} \approx -0.47140452079103184 \][/tex]
- For the negative square root:
[tex]\[ x_2 = \frac{-5 - 1}{6 \sqrt{2}} \][/tex]
[tex]\[ x_2 = \frac{-6}{6 \sqrt{2}} \][/tex]
[tex]\[ x_2 = \frac{-1}{\sqrt{2}} \][/tex]
Rationalizing the denominator, we get:
[tex]\[ x_2 = -\frac{\sqrt{2}}{2} \approx -0.7071067811865472 \][/tex]
### Summary of the solutions:
- Discriminant: [tex]\(\Delta = 1\)[/tex]
- Roots: [tex]\(x_1 \approx -0.47140452079103184\)[/tex] and [tex]\(x_2 \approx -0.7071067811865472\)[/tex]
Thus, the solutions to the quadratic equation [tex]\(3 \sqrt{2} x^2 + 5 x + \sqrt{2} = 0\)[/tex] are:
[tex]\[ x_1 \approx -0.47140452079103184 \][/tex]
[tex]\[ x_2 \approx -0.7071067811865472 \][/tex]
