To solve the quadratic equation [tex]\(2x^2 + x - 4 = 0\)[/tex], we will use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = -4 \)[/tex]
Firstly, we need to calculate the discriminant, which is given by [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{Discriminant} = 1^2 - 4 \cdot 2 \cdot (-4)
\][/tex]
[tex]\[
\text{Discriminant} = 1 + 32
\][/tex]
[tex]\[
\text{Discriminant} = 33
\][/tex]
The discriminant is 33.
Now, we apply these values into the quadratic formula to find the two solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-1 \pm \sqrt{33}}{2 \cdot 2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{33}}{4}
\][/tex]
This yields two solutions:
[tex]\[
x_1 = \frac{-1 + \sqrt{33}}{4}
\][/tex]
[tex]\[
x_2 = \frac{-1 - \sqrt{33}}{4}
\][/tex]
Approximating the values:
[tex]\[
x_1 \approx 1.186
\][/tex]
[tex]\[
x_2 \approx -1.686
\][/tex]
So, the solutions to the quadratic equation [tex]\(2x^2 + x - 4 = 0\)[/tex] are approximately:
[tex]\[
x_1 \approx 1.186
\][/tex]
[tex]\[
x_2 \approx -1.686
\][/tex]
