If one root of the quadratic equation $$ 6 x^{2}-k x+48=0 $$ is the square of the other, find the two roots and hence fing the value of $$ k $$.

Question

UpStudy AI · Accepted Answer

Let’s solve the quadratic equation:

$$ 6x^2 - kx + 48 = 0 $$

**Given:** One root is the square of the other. Let’s denote the roots as $$ \alpha $$ and $$ \alpha^2 $$.

For a quadratic equation $$ ax^2 + bx + c = 0 $$:

- **Sum of roots:** $$ \alpha + \alpha^2 = \frac{k}{6} $$
- **Product of roots:** $$ \alpha \cdot \alpha^2 = \frac{48}{6} = 8 $$

From the product:

$$ \alpha^3 = 8 \implies \alpha = 2 $$

(We consider only the real solution here, as complex roots would not satisfy the real coefficients of the quadratic equation.)

Now, the other root is:

$$ \alpha^2 = 2^2 = 4 $$

**Thus, the roots are 2 and 4.**

To find $$ k $$, use the sum of the roots:

$$ 2 + 4 = \frac{k}{6} \implies 6 = \frac{k}{6} \implies k = 36 $$

**Final Answer:**
The equation has roots 2 and 4, and the value of $$ k $$ is 36.
The roots are 2 and 4, and $$ k = 36 $$.

If one root of the quadratic equation \( 6 x^{2}-k x+48=0 \) is the square of the other, find the two roots and hence fing the value of \( k \).