Question

Write down the quadratic equation whose roots are 2 and -3.

Hint:

To form a quadratic equation from given roots, use the formula \( x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \).

Answer 1

Step 1: Write the given roots.}
The given roots of the quadratic equation are \( 2 \) and \( -3 \).
Step 2: Use the standard form of equation from roots.}
If \( \alpha \) and \( \beta \) are the roots of a quadratic equation, then the equation is given by \[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Step 3: Find the sum and product of roots.}
Here, \[ \alpha = 2, \quad \beta = -3 \] So, the sum of the roots is \[ \alpha + \beta = 2 + (-3) = -1 \] and the product of the roots is \[ \alpha \beta = 2 \times (-3) = -6 \]
Step 4: Substitute in the formula.}
Putting these values in the standard equation, we get \[ x^2 - (-1)x + (-6) = 0 \]
Step 5: Simplify the equation.}
On simplification, the required quadratic equation becomes \[ x^2 + x - 6 = 0 \]