Sure, let's solve the quadratic equation [tex]\(\sqrt{2} x^2 + 6 x + 4 \sqrt{2} = 0\)[/tex] step-by-step.
1. Identify the coefficients:
The given quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
- [tex]\(a = \sqrt{2}\)[/tex]
- [tex]\(b = 6\)[/tex]
- [tex]\(c = 4\sqrt{2}\)[/tex]
2. Quadratic Formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is found using [tex]\(b^2 - 4ac\)[/tex].
[tex]\[
\Delta = 6^2 - 4 \cdot \sqrt{2} \cdot 4\sqrt{2}
\][/tex]
Simplify the discriminant:
[tex]\[
\Delta = 36 - 4 \cdot \sqrt{2} \cdot 4\sqrt{2}
\][/tex]
[tex]\[
\Delta = 36 - 4 \cdot (2) \cdot 4
\][/tex]
[tex]\[
\Delta = 36 - 32
\][/tex]
[tex]\[
\Delta = 4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-6 \pm \sqrt{4}}{2 \cdot \sqrt{2}}
\][/tex]
Simplify the expression under the square root and solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-6 \pm 2}{2\sqrt{2}}
\][/tex]
This gives us two solutions to solve separately:
Solution 1:
[tex]\[
x_1 = \frac{-6 + 2}{2\sqrt{2}} = \frac{-4}{2\sqrt{2}}
\][/tex]
Simplify:
[tex]\[
x_1 = \frac{-4}{2\sqrt{2}} = \frac{-4}{2} \cdot \frac{1}{\sqrt{2}} = -2 \cdot \frac{1}{\sqrt{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}
\][/tex]
Solution 2:
[tex]\[
x_2 = \frac{-6 - 2}{2\sqrt{2}} = \frac{-8}{2\sqrt{2}}
\][/tex]
Simplify:
[tex]\[
x_2 = \frac{-8}{2\sqrt{2}} = \frac{-8}{2} \cdot \frac{1}{\sqrt{2}} = -4 \cdot \frac{1}{\sqrt{2}} = -\frac{4}{\sqrt{2}} = -2\sqrt{2}
\][/tex]
5. Present the Solutions:
The solutions to the equation [tex]\(\sqrt{2} x^2 + 6 x + 4 \sqrt{2} = 0\)[/tex] are:
[tex]\[
x = -2\sqrt{2} \quad \text{and} \quad x = -\sqrt{2}
\][/tex]
