To find the zeroes of the polynomial [tex]\( f(x) = 2x^2 + 7x - 4 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the polynomial [tex]\( ax^2 + bx + c \)[/tex].
For our polynomial [tex]\( 2x^2 + 7x - 4 \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = -4 \)[/tex]
Step 1: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 7^2 - 4 \cdot 2 \cdot (-4)
\][/tex]
[tex]\[
\Delta = 49 + 32
\][/tex]
[tex]\[
\Delta = 81
\][/tex]
Step 2: Calculate the roots using the quadratic formula
Since the discriminant [tex]\( \Delta \)[/tex] is positive, we have two distinct real roots. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
[tex]\[
x = \frac{-7 \pm \sqrt{81}}{4}
\][/tex]
[tex]\[
x = \frac{-7 \pm 9}{4}
\][/tex]
Now, compute the two possible values for [tex]\( x \)[/tex]:
Root 1:
[tex]\[
x_1 = \frac{-7 + 9}{4}
\][/tex]
[tex]\[
x_1 = \frac{2}{4}
\][/tex]
[tex]\[
x_1 = 0.5
\][/tex]
Root 2:
[tex]\[
x_2 = \frac{-7 - 9}{4}
\][/tex]
[tex]\[
x_2 = \frac{-16}{4}
\][/tex]
[tex]\[
x_2 = -4.0
\][/tex]
Thus, the zeroes of the polynomial [tex]\( 2x^2 + 7x - 4 \)[/tex] are:
[tex]\[
x_1 = 0.5 \quad \text{and} \quad x_2 = -4.0
\][/tex]
So, one of the zeroes of the polynomial [tex]\( f(x) = 2x^2 + 7x - 4 \)[/tex] is:
[tex]\[
\boxed{0.5}
\][/tex]
or alternatively,
[tex]\[
\boxed{-4.0}
\][/tex]
