To solve the quadratic equation [tex]\( 2x^2 + x - 4 = 0 \)[/tex], we can proceed by using the quadratic formula. The quadratic formula is given by:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}
\][/tex]
Here, the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = -4 \)[/tex]
First, we calculate the discriminant, which is [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substitute 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]\[
\text{Discriminant} = 1 + 32
\][/tex]
[tex]\[
\text{Discriminant} = 33
\][/tex]
Now that we have the discriminant, we can find the solutions to the equation using the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{\text{Discriminant}}}}}{{2a}}
\][/tex]
[tex]\[
x = \frac{{-1 \pm \sqrt{33}}}{{2 \cdot 2}}
\][/tex]
[tex]\[
x = \frac{{-1 \pm \sqrt{33}}}{4}
\][/tex]
This gives us two solutions:
1.
[tex]\[
x_1 = \frac{{-1 + \sqrt{33}}}{4}
\][/tex]
2.
[tex]\[
x_2 = \frac{{-1 - \sqrt{33}}}{4}
\][/tex]
Evaluating these expressions, we get the approximate numerical solutions:
[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] and [tex]\( x_2 \approx -1.686 \)[/tex].
