CBSE · Class 10 · Mathematics · PolynomialsFind the zeros of the quadratic polynomial $x^2 + 7x + 10$, and verify the relationship between the zeros and the coefficients.

To find the zeros of $p(x) = x^2 + 7x + 10$, we use the splitting the middle term method:

1. Factorization: $x^2 + 7x + 10 = 0$ $x^2 + 5x + 2x + 10 = 0$ $x(x + 5) + 2(x + 5) = 0$ $(x + 2)(x + 5) = 0$

   So, $x + 2 = 0$ or $x + 5 = 0$ $x = -2$ or $x = -5$ Therefore, the zeros are $\alpha = -2$ and $\beta = -5$.

2. Verification: Comparing $x^2 + 7x + 10$ with $ax^2 + bx + c$, we get $a = 1, b = 7, c = 10$.

   • Sum of zeros: $\alpha + \beta = (-2) + (-5) = -7$ Formula: $-\frac{b}{a} = -\frac{7}{1} = -7$ Hence, Sum of zeros $= -\frac{b}{a}$ (Verified).

   • Product of zeros: $\alpha \cdot \beta = (-2) \times (-5) = 10$ Formula: $\frac{c}{a} = \frac{10}{1} = 10$ Hence, Product of zeros $= \frac{c}{a}$ (Verified).