Question

The equation \[7{x^2} - \left( {7\pi + 22} \right)x + 22\pi = 0\] has
1) equal roots
2) a root which is negative
3) rational roots
4) a rational root and an irrational root

Answer 1

Hint: Firstly, compare the given equation with the standard quadratic equation, \[A{x^2} + Bx + C = 0\]. Then substitute the value of \[A = 7\], \[B = - \left( {7\pi + 22} \right)\] and \[C = 22\pi \]in the formula of discriminant, \[D = {B^2} - 4AC\]. If the value of \[D\] is 0, the equation has equal roots. If the value of \[D\] is greater than 0, the equation has two real roots and if the value of \[D\] is less than 0, the equation has no roots. Further, the roots can be calculated by using the formula, \[\dfrac{{ - B \pm \sqrt D }}{{2A}}\]

Complete step by step solution:
Firstly, compare the given equation with the standard quadratic equation, \[A{x^2} + Bx + C = 0\].
We are given that the equation is \[7{x^2} - \left( {7\pi + 22} \right)x + 22\pi = 0\]
Here, we see that \[A = 7\], \[B = - \left( {7\pi + 22} \right)\] and \[C = 22\pi \]
To find the number and type of roots, find the discriminant of the given equation.
The discriminant for the equation, \[A{x^2} + Bx + C = 0\] is given by \[D = {B^2} - 4AC\]
Substitute the values of \[A = 7\], \[B = - \left( {7\pi + 22} \right)\] and \[C = 22\pi \] in the formula of discriminant.
Thus, discriminant for the given equation can be obtained as,
\[D = {\left( { - \left( {7\pi + 22} \right)} \right)^2} - 4\left( 7 \right)\left( {22\pi } \right)\]
Open the bracket using the formula, \[{\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy\]
$
  D = {\left( { - \left( {7\pi + 22} \right)} \right)^2} - 4\left( 7 \right)\left( {22\pi } \right) \\
  D = 49{\pi ^2} + 484 + 308\pi - 616\pi \\
  D = 49{\pi ^2} + 484 - 308\pi \\
$
Use \[\dfrac{{22}}{7}\] for \[\pi \] in the above expression.
$
  D = 49{\left( {\dfrac{{22}}{7}} \right)^2} + 484 - 308\left( {\dfrac{{22}}{7}} \right) \\
  \Rightarrow D = 49\dfrac{{{{\left( {22} \right)}^2}}}{{{7^2}}} + 484 - 308\left( {\dfrac{{22}}{7}} \right) \\
  \Rightarrow D = 49\dfrac{{{{\left( {22} \right)}^2}}}{{49}} + 484 - 308\left( {\dfrac{{22}}{7}} \right) \\
  \Rightarrow D = {\left( {22} \right)^2} + 484 - 44\left( {22} \right) \\
  \Rightarrow D = 484 + 484 - 44\left( {22} \right) \\
  \Rightarrow D = 968 - 968 \\
  \Rightarrow D = 0 \\
$
Since the value of the discriminant is equal to 0, the quadratic equation has equal roots.
Now, we will found the roots of the equation \[7{x^2} - \left( {7\pi + 22} \right)x + 22\pi = 0\]
The roots can be calculated using the formula, \[\dfrac{{ - B \pm \sqrt D }}{{2A}}\]
On substituting the value \[A = 7\], \[B = - \left( {7\pi + 22} \right)\] and \[C = 22\pi \]in \[\dfrac{{ - B \pm \sqrt D }}{{2A}}\], we get,
$
  \dfrac{{ - B \pm \sqrt D }}{{2A}} = \dfrac{{7\pi + 22 \pm 0}}{{2\left( 7 \right)}} \\
   = \dfrac{{7\left( {\dfrac{{22}}{7}} \right) + 22 \pm 0}}{{2\left( 7 \right)}} \\
$
On solving the brackets, we get,
$
   = \dfrac{{22 + 22}}{{14}} \\
   = \dfrac{{44}}{{14}} \\
   = \dfrac{{22}}{7} \\
   = \pi \\
$
Hence, option (1) is the correct option.

Note: This question can also be solved by dividing on both sides by 7. On dividing we will get, \[{x^2} - \left( {\pi + \dfrac{{22}}{7}} \right)x + \left( {\dfrac{{22}}{7}} \right)\pi = 0\]. Then substituting the value of \[\dfrac{{22}}{7}\] as \[\pi \]. The equation will then be written as, \[{x^2} - \left( {2\pi } \right)x + {\pi ^2} = 0\]which is equal to \[{\left( {x - \pi } \right)^2} = 0\]. Thus, the equation has equal roots, \[x = \pi ,\pi \]. Also, \[\pi \] is an irrational number.