To determine the number of real-number solutions for the given quadratic equation [tex]\(x^2 + 2x - 4 = 0\)[/tex], we need to use the discriminant. The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the equation. For the equation [tex]\(x^2 + 2x - 4 = 0\)[/tex], the coefficients are:
[tex]\[
a = 1, \quad b = 2, \quad c = -4
\][/tex]
Now, we substitute these values into the discriminant formula:
[tex]\[
\Delta = 2^2 - 4 \cdot 1 \cdot (-4)
\][/tex]
Calculating this step-by-step:
1. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
2. Calculate [tex]\(4ac\)[/tex]:
[tex]\[
4 \cdot 1 \cdot (-4) = -16
\][/tex]
3. Combine these results:
[tex]\[
\Delta = 4 - (-16) = 4 + 16 = 20
\][/tex]
The discriminant [tex]\(\Delta\)[/tex] is 20.
Next, we interpret the discriminant to determine the number of real-number solutions:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real-number solutions.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real-number solution.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real-number solutions.
In this case, [tex]\(\Delta = 20\)[/tex], which is greater than 0. Therefore, the equation [tex]\(x^2 + 2x - 4 = 0\)[/tex] has two distinct real-number solutions.
The correct answer is:
A. The equation has two real-number solutions.
