To solve the given quadratic equation [tex]\(7x^2 + 10x + 3 = 0\)[/tex], we need to follow these steps:
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 7\)[/tex]
- [tex]\(b = 10\)[/tex]
- [tex]\(c = 3\)[/tex]
2. Calculate the discriminant ([tex]\(D\)[/tex]) using the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
3. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
D = (10)^2 - 4(7)(3)
\][/tex]
4. Perform the calculations:
[tex]\[
10^2 = 100
\][/tex]
[tex]\[
4 \cdot 7 = 28
\][/tex]
[tex]\[
28 \cdot 3 = 84
\][/tex]
[tex]\[
D = 100 - 84 = 16
\][/tex]
So, the discriminant [tex]\(D\)[/tex] is:
[tex]\[
D = 16
\][/tex]
Next, we interpret the value of the discriminant:
5. If [tex]\(D > 0\)[/tex], the quadratic equation has two distinct real solutions.
6. If [tex]\(D = 0\)[/tex], the quadratic equation has one real repeated solution.
7. If [tex]\(D < 0\)[/tex], the quadratic equation has two complex solutions.
Since [tex]\(D = 16\)[/tex] ([tex]\(D > 0\)[/tex]), the quadratic equation [tex]\(7x^2 + 10x + 3 = 0\)[/tex] has two real and distinct solutions.
Therefore, the final result is:
The discriminant [tex]\(D = 16\)[/tex]. There are 2 real and distinct solutions to the quadratic equation with this discriminant.
