Question

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

Answer 1

Hint: We need to convert the given quadratic equation to its standard form $a{x^2} + bx + c = 0$. Then we can find the discriminant using the equation $D = {b^2} - 4ac$ and thus find the type of roots of the equation. If D is positive the roots will be real and distinct, for D=0, the equation has only 1 root and if D is negative, the equation will have no real root.

Complete step by step Answer:

The given equation is a quadratic equation. It is given in the factorized form, we must convert to standard form.
$
  \left( {{\text{x - 705}}} \right)\left( {{\text{x - 795}}} \right){\text{ + 800}}\left( {{\text{x - 750}}} \right)\left( {{\text{x - 835}}} \right){\text{ = 0}} \\
  {\text{On simplification we get,}} \\
  {{\text{x}}^{\text{2}}}{\text{ - }}\left( {{\text{705 + 795}}} \right){\text{x + 705 $\times$ 795 + 800}}\left( {{{\text{x}}^{\text{2}}}{\text{ - }}\left( {{\text{750 + 835}}} \right){\text{x + 750 $\times$ 835}}} \right){\text{ = 0}} \\
  {\text{On taking common terms together we get,}} \\
  {\text{801}}{{\text{x}}^{\text{2}}}{\text{ - }}\left( {{\text{1500 + 800 $\times$ 1585}}} \right){\text{x + 560475 $\times$ 800 $\times$ 626250 = 0}} \\
  {\text{801}}{{\text{x}}^{\text{2}}}{\text{ - 1269500x + 280797975000000 = 0}} \\
 $
 Now we have the equation in standard form. We can find the nature of roots from the discriminant.
Discriminant is given by ${\text{D = }}{{\text{b}}^{\text{2}}}{\text{ - 4ac}}$
$
  {\text{D = }}{\left( {{\text{ - 1269500}}} \right)^{\text{2}}}{\text{ - 4 $\times$ 801 $\times$ 280,797,975,000,000}} \\
  {\text{ = 1,611,630,250,000 - 3024 $\times$ 280,797,975,000,000}} \\
 $
It is clear from the digits that the discriminant is a negative value. So, there will be no real roots.
The equation will only have imaginary roots.
Therefore, the correct answer is option A.

Note: The nature of the roots of a quadratic equation can be determined by calculating the discriminant D.
 If D>0, the equation will have real and distinct roots.
If D=0, the equation will have equal roots.
If D<0, the equation has only imaginary roots.
The value of D can be found for quadratic equations in its standard form. In this problem, we need not find the exact value of the discriminant. As we are not trying to find the solution, we only need to know whether the discriminant is positive or negative or zero. Equations having no real roots will have complex roots in terms of i where ${i^2} = - 1$. For equations with positive discriminant, the roots are given by the equation, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.