Formation of a Quadratic Equation with Given Roots

Therefore, the required quadratic equation is:

\[ \boxed{x^{2} - (\alpha + \beta)x + \alpha\beta = 0} \]

Remember:

Coefficient of \[ x \] = \[ -(\text{sum of roots}) \]

Constant term = \[ +(\text{product of roots}) \]

For roots \[ \alpha \] and \[ \beta: \]

\[ \boxed{x^{2} - (\alpha + \beta)x + \alpha\beta = 0} \]

If the equation is not required to be monic, then for any non-zero constant \[ k: \]

\[ \boxed{k\left[x^{2} - (\alpha + \beta)x + \alpha\beta\right] = 0} \]

All such equations have the same roots \[ \alpha \] and \[ \beta. \]

Activity (I)

Given: Addition of the roots = \[ 10, \] Product of the roots = \[ 9 \]

Completed Equation:

\[ \therefore\; \text{Quadratic equation} : x^{2} - 10x + 9 = 0 \]

Activity (II)

Given: \[ \alpha = 2, \beta = 5 \]

Completed Steps:

It can be written as \[ x^{2} - (2 + 5)x + 2 \times 5 = 0. \]

That is \[ 1x^{2} - 7x + 10 = 0. \]

• If roots are \[ \alpha \] and \[ \beta, \] the equation is \[ x^{2} - (\alpha + \beta)x + \alpha\beta = 0. \]
• The middle-term coefficient is always negative of the sum of roots.
• The constant term is the product of roots.
• Multiplying the whole equation by any non-zero constant keeps the roots unchanged.