Question

Find the​ x-intercept(s) algebraically.
​f(x)equals2 x squared minus 11 x plus 14
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Part 1
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Answer 1

To find the x-intercepts of the function \( f(x) = 2x^2 - 11x + 14 \), we need to solve the equation \( f(x) = 0 \).

This means we need to set the equation equal to zero and solve for \( x \):

\[
2x^2 - 11x + 14 = 0
\]

Now, we can use the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 2 \), \( b = -11 \), and \( c = 14 \).

1. Calculate the discriminant:

\[
b^2 - 4ac = (-11)^2 - 4 \cdot 2 \cdot 14
\]
\[
= 121 - 112 = 9
\]

2. Now substitute into the quadratic formula:

\[
x = \frac{-(-11) \pm \sqrt{9}}{2 \cdot 2}
\]
\[
x = \frac{11 \pm 3}{4}
\]

3. This gives us two possible solutions:

- For the positive case:

\[
x = \frac{11 + 3}{4} = \frac{14}{4} = 3.5
\]

- For the negative case:

\[
x = \frac{11 - 3}{4} = \frac{8}{4} = 2
\]

Thus, the x-intercepts of the function \( f(x) = 2x^2 - 11x + 14 \) are:

\[
x = 2 \quad \text{and} \quad x = 3.5
\]

So the x-intercepts are \( x = 2 \) and \( x = 3.5 \).