To determine the number of roots for the quadratic equation [tex]\(8x^2 + 7x - 19 = 0\)[/tex], we need to calculate the discriminant.
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants. For the given equation:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 7\)[/tex]
- [tex]\(c = -19\)[/tex]
The discriminant ([tex]\(\Delta\)[/tex]) is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 7^2 - 4(8)(-19) \][/tex]
First, calculate the values inside the parenthesis and the square:
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 4 \times 8 = 32 \][/tex]
[tex]\[ 32 \times (-19) = -608 \][/tex]
Notice, the negative sign changes when multiplying two negative numbers:
[tex]\[ -608 \rightarrow +608 \][/tex]
Now, substitute back into the discriminant formula:
[tex]\[ \Delta = 49 + 608 \][/tex]
[tex]\[ \Delta = 657 \][/tex]
With [tex]\(\Delta = 657\)[/tex], we can determine the number of roots based on the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has 2 distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly 1 real root.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real roots (the roots are complex).
Since [tex]\(\Delta = 657\)[/tex] is greater than 0, the quadratic equation has 2 distinct real roots.
Therefore, the equation [tex]\(8x^2 + 7x - 19 = 0\)[/tex] has 2 roots.
The answer is:
2
