Question

Question

What are the zeroes of the function f left parenthesis x right parenthesis equals 2 x squared plus 5 x minus 3?
(1 point)
Responses

x equals negative one half semicolon x equals 3
Image with alt text: x equals negative one half semicolon x equals 3

x equals start fraction 3 over 2 end fraction semicolon x equals negative 1
Image with alt text: x equals start fraction 3 over 2 end fraction semicolon x equals negative 1

x equals negative start fraction 3 over 2 semicolon x equals 1
Image with alt text: x equals negative start fraction 3 over 2 semicolon x equals 1

x equals one half semicolon x equals negative 3
Image with alt text: x equals one half semicolon x equals negative 3
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Answer 1

To find the zeroes of the function \( f(x) = 2x^2 + 5x - 3 \), we need to solve the equation \( 2x^2 + 5x - 3 = 0 \).

We can use the quadratic formula:

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

Here, \( a = 2 \), \( b = 5 \), and \( c = -3 \).

First, we calculate the discriminant:

\[ b^2 - 4ac = 5^2 - 4(2)(-3) = 25 + 24 = 49 \]

Now, we can use the quadratic formula:

\[ x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4} \]

Calculating the two potential solutions:

1. \( x = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2} \)

2. \( x = \frac{-5 - 7}{4} = \frac{-12}{4} = -3 \)

Thus, the roots (or zeroes) of the function are:

\[ x = \frac{1}{2}, \quad x = -3 \]

Based on the responses provided, the correct answer is:

x equals one half; x equals negative 3.