Question

Find the zeroes of the quadratic polynomial $x^2 + 7x + 10$ and verify the relationship between the zeroes and the coefficients.

Hint:

Always check: Sum of zeroes = $-\dfrac{b}{a}$ and Product of zeroes = $\dfrac{c}{a}$.

Answer 1

Step 1: Factorize the polynomial.
\[ x^2 + 7x + 10 = 0 \] \[ x^2 + 5x + 2x + 10 = 0 \Rightarrow x(x + 5) + 2(x + 5) = 0 \Rightarrow (x + 5)(x + 2) = 0 \]
Step 2: Find the zeroes.
\[ x + 5 = 0 \Rightarrow x = -5, \quad x + 2 = 0 \Rightarrow x = -2 \]
Step 3: Verify the relationship.
Sum of zeroes \( = (-5) + (-2) = -7 \) Product of zeroes \( = (-5)(-2) = 10 \) From the polynomial \( ax^2 + bx + c \), \[ \text{Sum of zeroes} = -\dfrac{b}{a} = -\dfrac{7}{1} = -7, \quad \text{Product of zeroes} = \dfrac{c}{a} = \dfrac{10}{1} = 10 \] Both relations are verified.

Step 4: Conclusion.
Hence, the zeroes are -5 and -2, and the relationships between zeroes and coefficients are verified.