Determine the discriminant for the quadratic equation [tex]-3 = x^2 + 4x + 1[/tex]. Based on the discriminant value, how many real number solutions does the equation have?

Discriminant: [tex]b^2 - 4ac[/tex]

A. 0
B. 1
C. 2
D. 12



Answer :

To solve the quadratic equation [tex]\(-3 = x^2 + 4x + 1\)[/tex], we first need to rewrite it in standard quadratic form:

[tex]\[ x^2 + 4x + 1 + 3 = 0 \][/tex]

Simplifying the constants on the left side gives us:

[tex]\[ x^2 + 4x + 4 = 0 \][/tex]

In the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
[tex]\( a = 1 \)[/tex],
[tex]\( b = 4 \)[/tex],
[tex]\( c = 4 \)[/tex].

We calculate the discriminant using the formula:

[tex]\[ \Delta = b^2 - 4ac \][/tex]

Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:

[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0 \][/tex]

The discriminant [tex]\(\Delta\)[/tex] is 0.

Based on the value of the discriminant, we determine the number of real number solutions for the quadratic equation:
- If the discriminant is greater than 0, there are 2 distinct real solutions.
- If the discriminant is 0, there is exactly 1 real solution.
- If the discriminant is less than 0, there are no real solutions.

Since the discriminant is 0, the quadratic equation has exactly 1 real solution.

So, the discriminant is 0, and there is 1 real number solution for the equation [tex]\(-3 = x^2 + 4x + 1\)[/tex].