Certainly! Let's solve the quadratic equation [tex]\( y = 2x^2 - 4x - 2 \)[/tex] step by step to find the roots, or the values of [tex]\( x \)[/tex] where [tex]\( y \)[/tex] is equal to zero.
1. Write the quadratic equation in standard form:
[tex]\[
2x^2 - 4x - 2 = 0
\][/tex]
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = -2 \)[/tex]
3. Use the quadratic formula to find the roots:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}
\][/tex]
4. Simplify inside the square root:
[tex]\[
x = \frac{4 \pm \sqrt{16 + 16}}{4}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{32}}{4}
\][/tex]
5. Simplify the square root:
[tex]\[
\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}
\][/tex]
6. Substitute back into the equation:
[tex]\[
x = \frac{4 \pm 4\sqrt{2}}{4}
\][/tex]
7. Simplify the fraction:
[tex]\[
x = 1 \pm \sqrt{2}
\][/tex]
So the solutions (roots) of the quadratic equation [tex]\( y = 2x^2 - 4x - 2 \)[/tex] are:
[tex]\[
x = 1 - \sqrt{2} \quad \text{and} \quad x = 1 + \sqrt{2}
\][/tex]
These are the points at which the graph of the quadratic equation intersects the x-axis.
