Question

The equation $\left( {x - 705} \right)\left( {x - 795} \right) + 800\left( {x - 750} \right)\left( {x - 835} \right) = 0$ has
A. imaginary roots
B. Equal roots
C. Distinct real roots
D. None of these

Answer 1

Hint:
We will simplify the equation and will write in the form $a{x^2} + bx + c = 0$. Then, we will calculate the value of discriminant, using the formula, $D = {b^2} - 4ac$ . The value of $D$ will help us to find the type of roots. If it is positive , then the equation has real roots, if the value of $D$ is negative, then the roots are imaginary and if $D = 0$, the equation has equal roots.

Complete step by step solution:
We are given the equation $\left( {x - 705} \right)\left( {x - 795} \right) + 800\left( {x - 750} \right)\left( {x - 835} \right) = 0$
We will multiply the brackets and then simplify the equation.
${x^2} - \left( {705 + 795} \right)x + 705\left( {795} \right) + 800\left( {{x^2} - \left( {750 + 835} \right)x + 750\left( {835} \right)} \right) = 0$
When we will take common terms together, we will get,
$
  801{x^2} - \left( {1500 + 800\left( {1585} \right)} \right)x + 705\left( {795} \right) + 800\left( {560475} \right)\left( {626250} \right) = 0 \\
   \Rightarrow 801{x^2} - 1269500x + 280797975000000 = 0 \\
$
The above equation is of the form, $a{x^2} + bx + c = 0$.
We can now calculate the value of discriminant using the formula, $D = {b^2} - 4ac$
$
  D = {\left( { - 1269500} \right)^2} - 4\left( {81} \right)\left( {280,797,975,000,000} \right) \\
   = 1,611,630,250,000 - 3024\left( {280,797,975,000,000} \right) \\
$
The above value will be less than 0.
Hence, the roots of the given equation are imaginary.

Thus, option A is correct.

Note:
We determine the nature of the roots by finding the value of discriminant, $D = {b^2} - 4ac$, where the quadratic equation is of the form $a{x^2} + bx + c = 0$.
1) If $D > 0$, then the equation will have real roots.
2) If $D = 0$, then the equation has real and distinct roots.
3) And if $D < 0$, then the equation has no real roots.