To find the x-intercepts for the parabola defined by the equation [tex]\( y = 2x^2 - x - 10 \)[/tex], we can use the quadratic formula: [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex].
Given the quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], we have:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = -1 \][/tex]
[tex]\[ c = -10 \][/tex]
1. Calculate the Discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substituting the values:
[tex]\[
\text{Discriminant} = (-1)^2 - 4 \cdot 2 \cdot (-10) = 1 + 80 = 81
\][/tex]
2. Calculate the Square Root of the Discriminant:
[tex]\[
\sqrt{81} = 9
\][/tex]
3. Apply the Quadratic Formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting [tex]\( b = -1 \)[/tex], [tex]\(\sqrt{b^2 - 4ac} = 9\)[/tex], and [tex]\( a = 2 \)[/tex]:
[tex]\[
x_1 = \frac{-(-1) + 9}{2 \cdot 2} = \frac{1 + 9}{4} = \frac{10}{4} = 2.5
\][/tex]
[tex]\[
x_2 = \frac{-(-1) - 9}{2 \cdot 2} = \frac{1 - 9}{4} = \frac{-8}{4} = -2.0
\][/tex]
4. Determine the Correct Intercept:
One of the x-intercepts is given as [tex]\((2.5, 0)\)[/tex]. Thus, the other x-intercept calculated must be the correct one.
So, the other [tex]\( x \)[/tex]-intercept is:
[tex]\[
\boxed{2.5, -2.00}
\][/tex]
Given that one [tex]\( x \)[/tex]-intercept is [tex]\( (2.5, 0) \)[/tex], the other [tex]\( x \)[/tex]-intercept for the equation [tex]\( y = 2x^2 - x - 10 \)[/tex] is:
[tex]\[
-2.00
\][/tex]
