Answer :
Certainly! To find the value of [tex]\( k \)[/tex] in the given quadratic equation [tex]\( 5x^2 + 13x + k = 0 \)[/tex] under the condition that one root is the reciprocal of the other root, we follow these steps:
### Step 1: Understand the roots relationship
Suppose the roots of the quadratic equation are [tex]\( p \)[/tex] and [tex]\( \frac{1}{p} \)[/tex]. Given that one root is the reciprocal of the other, we can utilize the relationships formed by the roots of a quadratic equation.
### Step 2: Use the product of roots
The product of the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \text{Product of the roots} = \frac{c}{a} \][/tex]
For our specific equation [tex]\( 5x^2 + 13x + k = 0 \)[/tex], we have:
[tex]\[ \text{Product of the roots} = p \cdot \frac{1}{p} = 1 \][/tex]
Thus:
[tex]\[ \frac{k}{5} = 1 \][/tex]
### Step 3: Solve for [tex]\( k \)[/tex]
To find [tex]\( k \)[/tex], we solve the equation:
[tex]\[ \frac{k}{5} = 1 \implies k = 5 \][/tex]
### Conclusion
Therefore, the value of [tex]\( k \)[/tex] for which one root of the equation [tex]\( 5x^2 + 13x + k = 0 \)[/tex] is the reciprocal of the other root is:
[tex]\[ k = 5 \][/tex]
### Step 1: Understand the roots relationship
Suppose the roots of the quadratic equation are [tex]\( p \)[/tex] and [tex]\( \frac{1}{p} \)[/tex]. Given that one root is the reciprocal of the other, we can utilize the relationships formed by the roots of a quadratic equation.
### Step 2: Use the product of roots
The product of the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \text{Product of the roots} = \frac{c}{a} \][/tex]
For our specific equation [tex]\( 5x^2 + 13x + k = 0 \)[/tex], we have:
[tex]\[ \text{Product of the roots} = p \cdot \frac{1}{p} = 1 \][/tex]
Thus:
[tex]\[ \frac{k}{5} = 1 \][/tex]
### Step 3: Solve for [tex]\( k \)[/tex]
To find [tex]\( k \)[/tex], we solve the equation:
[tex]\[ \frac{k}{5} = 1 \implies k = 5 \][/tex]
### Conclusion
Therefore, the value of [tex]\( k \)[/tex] for which one root of the equation [tex]\( 5x^2 + 13x + k = 0 \)[/tex] is the reciprocal of the other root is:
[tex]\[ k = 5 \][/tex]
