To determine the values of [tex]\( m \)[/tex] and [tex]\( n \)[/tex] for the quadratic equation [tex]\( f(x) = x^2 + mx + n \)[/tex] given that its roots are 5 and -3, we can use properties of quadratic equations related to their roots.
### Step-by-Step Solution:
(a) Calculate the value of
(i) [tex]\( m \)[/tex]:
Step 1: Recall the sum of the roots of a quadratic equation.
For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the sum of the roots ([tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex]) is given by:
[tex]\[ r_1 + r_2 = -\frac{b}{a} \][/tex]
Step 2: Apply it to the given equation [tex]\( x^2 + mx + n = 0 \)[/tex].
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = m \)[/tex], and [tex]\( c = n \)[/tex].
Given the roots are 5 and -3, the sum of the roots is:
[tex]\[ 5 + (-3) = 2 \][/tex]
Therefore, for the given quadratic equation,
[tex]\[ -m = 2 \][/tex]
Step 3: Solve for [tex]\( m \)[/tex]:
[tex]\[ m = -2 \][/tex]
(ii) [tex]\( n \)[/tex]:
Step 1: Recall the product of the roots of a quadratic equation.
For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the product of the roots ([tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex]) is given by:
[tex]\[ r_1 \cdot r_2 = \frac{c}{a} \][/tex]
Step 2: Apply it to the given equation [tex]\( x^2 + mx + n = 0 \)[/tex].
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = m \)[/tex], and [tex]\( c = n \)[/tex].
Given the roots are 5 and -3, the product of the roots is:
[tex]\[ 5 \cdot (-3) = -15 \][/tex]
Therefore, for the given quadratic equation,
[tex]\[ n = -15 \][/tex]
### Final Answer:
(i) The value of [tex]\( m \)[/tex] is [tex]\( -2 \)[/tex].
(ii) The value of [tex]\( n \)[/tex] is [tex]\( -15 \)[/tex].
