To determine the value of [tex]\( P \)[/tex] in the quadratic equation [tex]\( x^2 + Px - 4 = 0 \)[/tex] given that one of the roots is -4, we can follow these steps:
1. Express the quadratic equation using the roots:
If one root is [tex]\( r_1 = -4 \)[/tex] and the other root is [tex]\( r_2 \)[/tex], then the quadratic equation can be expressed in the factored form as:
[tex]\[
(x - r_1)(x - r_2) = 0
\][/tex]
Substituting [tex]\( r_1 = -4 \)[/tex]:
[tex]\[
(x + 4)(x - r_2) = 0
\][/tex]
2. Expand the factored form:
Expanding the expression on the left-hand side yields:
[tex]\[
x^2 - r_2 x + 4x - 4 \times r_2 = 0 \implies x^2 + (4 - r_2)x - 4r_2 = 0
\][/tex]
3. Compare with the original equation:
Compare this with the original quadratic equation [tex]\( x^2 + Px - 4 = 0 \)[/tex]. We obtain two conditions:
[tex]\[
4 - r_2 = P \quad \text{(1)}
\][/tex]
[tex]\[
-4r_2 = -4 \quad \text{(2)}
\][/tex]
4. Solve for the other root:
From [tex]\( -4r_2 = -4 \)[/tex], we can solve for [tex]\( r_2 \)[/tex]:
[tex]\[
r_2 = 1
\][/tex]
5. Substitute [tex]\( r_2 \)[/tex] back into the equation to find [tex]\( P \)[/tex]:
Using [tex]\( r_2 = 1 \)[/tex] in the equation [tex]\( P = 4 - r_2 \)[/tex]:
[tex]\[
P = 4 - 1
\][/tex]
[tex]\[
P = 3
\][/tex]
Thus, the value of [tex]\( P \)[/tex] is [tex]\( \boxed{3} \)[/tex].
