To find the x-intercepts of the quadratic function [tex]\( f(x) = 6x^2 - 13x + 6 \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex]. The steps to solve this quadratic equation are as follows:
1. Identify the coefficients: First, let's identify the coefficients of the quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
- [tex]\( a = 6 \)[/tex]
- [tex]\( b = -13 \)[/tex]
- [tex]\( c = 6 \)[/tex]
2. Calculate the discriminant: The discriminant ([tex]\( \Delta \)[/tex]) of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-13)^2 - 4(6)(6)
\][/tex]
[tex]\[
\Delta = 169 - 144
\][/tex]
[tex]\[
\Delta = 25
\][/tex]
3. Determine the x-intercepts: Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex] to find the roots of the equation:
[tex]\[
x = \frac{-(-13) \pm \sqrt{25}}{2(6)}
\][/tex]
Simplify this to:
[tex]\[
x = \frac{13 \pm 5}{12}
\][/tex]
4. Calculate the two possible solutions:
- For the plus sign ([tex]\( + \)[/tex]):
[tex]\[
x_1 = \frac{13 + 5}{12} = \frac{18}{12} = 1.5
\][/tex]
- For the minus sign ([tex]\( - \)[/tex]):
[tex]\[
x_2 = \frac{13 - 5}{12} = \frac{8}{12} = \frac{2}{3} = 0.6666666666666666
\][/tex]
So, the x-intercepts of the quadratic function [tex]\( f(x) = 6x^2 - 13x + 6 \)[/tex] are [tex]\( x = 1.5 \)[/tex] and [tex]\( x = 0.6666666666666666 \)[/tex].
Thus, the x-intercepts are:
[tex]\[
x = 1.5 \quad \text{and} \quad x = 0.6666666666666666
\][/tex]
These are the points where the graph of the quadratic equation intersects the x-axis.
