Question

Determine the zero(s) of the quadratic function y = 2x^2 = 9x + 4.
(0,4)
(-4,0) (-0.5,0)
(4,0) (-1,0)
there are no zeros

Answer 1

To determine the zeros of the quadratic function given by the equation \( y = 2x^2 - 9x - 4 \), we need to set \( y \) equal to zero and solve for \( x \):

\[
2x^2 - 9x - 4 = 0
\]

We can use the quadratic formula, which is given by:

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

In this equation, \( a = 2 \), \( b = -9 \), and \( c = -4 \).

Now we can calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (-9)^2 - 4(2)(-4) = 81 + 32 = 113
\]

Since the discriminant is positive, there will be two real and distinct solutions.

Now we can substitute \( a \), \( b \), and the calculated discriminant into the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{113}}{2(2)} = \frac{9 \pm \sqrt{113}}{4}
\]

Thus, the solutions are:

\[
x = \frac{9 + \sqrt{113}}{4} \quad \text{and} \quad x = \frac{9 - \sqrt{113}}{4}
\]

Both solutions are real numbers. Therefore, the quadratic function \( y = 2x^2 - 9x - 4 \) has zeros, and the correct option from your list is: there are zero(s).