Solve [tex]\( 2x^2 + x - 4 = 0 \)[/tex]

Given:
[tex]\[ 2x^2 + x - 4 = 0 \][/tex]
[tex]\[ x^2 + \square x + \cdots = 0 \][/tex]

(Done)

(Note: the second equation "x^2 + \square x + -4" seems incomplete or incorrect as a quadratic equation. Ensure it makes sense in the context of the problem.)



Answer :

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].