Quadratic Equation from Sum and Product of Roots
Let the ages of Ishan and Anushka be the roots of a quadratic equation, denoted by '$r_1$' and '$r_2$'.
We are given the sum and product of their ages:
- Sum of ages: $r_1 + r_2 = 18$
- Product of ages: $r_1 \times r_2 = 56$
A standard quadratic equation can be formed using the sum and product of its roots using the formula:
$x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$
Substitute the given values into the formula:
$x^2 - (18)x + (56) = 0$
This simplifies to:
$x^2 - 18x + 56 = 0$
This equation represents the possible ages of Ishan and Anushka. Comparing this with the given options, we find that option 4 matches.
Age Calculation Verification
To verify, we can find the roots of the equation $x^2 - 18x + 56 = 0$. We look for two numbers that multiply to 56 and add up to 18. These numbers are 7 and 11? No, let's recheck. The numbers are 7 and 11, sum is 18, product is 77. Incorrect. Let's try 8 and 10? Sum is 18, product is 80. Incorrect. How about factoring 56? Factors are (1, 56), (2, 28), (4, 14), (7, 8). Let's check sums: 1+56=57, 2+28=30, 4+14=18, 7+8=15. Ah, the pair (4, 14) has a sum of 18 and a product of 56. So the ages are 4 and 14.
The equation is $x^2 - 18x + 56 = 0$. This matches option 4.
Important Questions from Identities
- If $x + y + z = 10$, $x^2 + y^2 + z^2 = 38$, $5xyz = 150$ and $(xy + yz + zx) = 31$, then what is the value of $2(x^3 + y^3 + z^3)$?
- If $x = \sqrt[3]{31 + 14\sqrt{3}}$ and $y = \sqrt[3]{31 - 14\sqrt{3}}$, then the value of $x^3 + y^3 + 3x^3y^3$ is
- If $8x^3 - 27y^3 = (Ax + By)(Cx^2 - Dy^2 + Exy)$, then the value of $(A + B + C + D + E)$ is:
- A mother has decided to sum and product the ages of her children Ishan and Anushka. The sum and product of their ages are 18 and 56, respectively. The solution of which quadratic equation gives the possible ages of Ishan and Anushka?
If $\frac{a+1}{a^2-1} = \frac{1}{5}$, then the value of $a$ is -
