Question

determine the discriminant for the quadratic equation -3 = x² + 4x + 1. based on the discriminant value, how many real number solutions does the equation have?
discriminant = b² - 4ac
○ 0
○ 1
○ 2
○ 12

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the quadratic equation \(-3 = x^{2}+4x + 1\) in the standard form \(ax^{2}+bx + c=0\).
Subtract \(-3\) from both sides (or add 3 to both sides) to get:
\(x^{2}+4x + 1+ 3=0\)
Simplify to:
\(x^{2}+4x+4 = 0\)
Here, \(a = 1\), \(b=4\), and \(c = 4\).

Step2: Calculate the discriminant

The formula for the discriminant \(D\) of a quadratic equation \(ax^{2}+bx + c = 0\) is \(D=b^{2}-4ac\).
Substitute \(a = 1\), \(b = 4\), and \(c=4\) into the formula:
\(D=(4)^{2}-4\times1\times4\)
First, calculate \((4)^{2}=16\) and \(4\times1\times4 = 16\).
Then, \(D=16 - 16\)
\(D = 0\)

Step3: Determine the number of real solutions

For a quadratic equation \(ax^{2}+bx + c=0\):

• If \(D>0\), there are 2 distinct real solutions.
• If \(D = 0\), there is 1 real solution (a repeated root).
• If \(D<0\), there are 0 real solutions.

Since \(D = 0\), the equation has 1 real solution.

Answer:

1 (corresponding to the option "1")