Let's solve the problem step by step.
### Step 1: Determine the value of [tex]\( p \)[/tex]
We know that [tex]\(-4\)[/tex] is a root of the quadratic equation:
[tex]\[ x^2 + px - 4 = 0 \][/tex]
A root of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] must satisfy it. Substituting [tex]\(-4\)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ (-4)^2 + p(-4) - 4 = 0 \][/tex]
This simplifies to:
[tex]\[ 16 - 4p - 4 = 0 \][/tex]
[tex]\[ 12 - 4p = 0 \][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[ 4p = 12 \][/tex]
[tex]\[ p = \frac{12}{4} \][/tex]
[tex]\[ p = 3 \][/tex]
### Step 2: Determine the value of [tex]\( q \)[/tex]
Next, we need to consider the second equation:
[tex]\[ x^2 + px + q = 0 \][/tex]
We are told that this equation has equal roots. For a quadratic equation with equal roots, the discriminant must be zero. The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the equation [tex]\( x^2 + px + q = 0 \)[/tex], we have [tex]\( a = 1 \)[/tex], [tex]\( b = p \)[/tex], and [tex]\( c = q \)[/tex]. Therefore, the discriminant is:
[tex]\[ \Delta = p^2 - 4 \cdot 1 \cdot q \][/tex]
[tex]\[ \Delta = p^2 - 4q \][/tex]
Since the equation has equal roots, the discriminant must be zero:
[tex]\[ p^2 - 4q = 0 \][/tex]
Substituting the known value of [tex]\( p \)[/tex]:
[tex]\[ (3)^2 - 4q = 0 \][/tex]
[tex]\[ 9 - 4q = 0 \][/tex]
Solving for [tex]\( q \)[/tex]:
[tex]\[ 4q = 9 \][/tex]
[tex]\[ q = \frac{9}{4} \][/tex]
### Conclusion
Therefore, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
[tex]\[ p = 3 \][/tex]
[tex]\[ q = \frac{9}{4} \][/tex]
