Question

If $8{x^2} - 22x - 21 = 0$ then $x = \dfrac{3}{4},\dfrac{5}{2}$
A.Yes
B.No
C.Ambiguous
D.Data Insufficient

Answer 1

Hint:- To find out the value of roots of a variable equation, we can use either Shri Dharacharya formula of factorisation method.


Complete Step by step by solution
We have given equation
$8{x^2} - 22x - 21 = 0$
Here, we have to each values of $x$; we have to find values of $x$
Every Quadratic equation has 2 roots; similarly every cubic equation has 3 roots.
Now, we will use the formula, which is as shown below –
If equation is $a{x^2} + bx + c = 0$
Then $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
According to question,
$
  a = 8, \\
  b = 22 \\
  c = 21 \\
$
$x = \dfrac{{ - 22 \pm \sqrt {{{( - 22)}^2} - 4 \times 8 \times ( - 21)} }}{{2 \times 8}}$
$x = \dfrac{{ - 22 \pm \sqrt {484 + 4 \times 8 \times 21} }}{{16}}$
$x = \dfrac{{ - 22 \pm \sqrt {484 + 672} }}{{16}}$
$x = \dfrac{{ - 22 \pm \sqrt {1156} }}{{16}}$
$x = \dfrac{{ - 22 \pm 34}}{{16}}$
To values of x are
$x = \left( {\dfrac{{22 + 34}}{{16}}} \right),\left( {\dfrac{{22 - 34}}{{16}}} \right)$
$x = \dfrac{{56}}{{16}},\dfrac{{ - 12}}{{16}}$
$x = \dfrac{7}{2},\dfrac{{ - 3}}{4}$
Hence one value of x is satisfied while another value is not so answer is ambiguous
i.e., C will be the right options.

Note –We can also check values of x using factorization method, as shown below,
Equation is
$8{x^2} - 22x - 21 = 0$
$ \Rightarrow 8{x^2} - 28x + 6x - 21 = 0$
$ \Rightarrow 4x(2x - 7) + 3(2x - 7) = 0$
$ \Rightarrow (2x - 7)(4x + 3) = 0$
$ \Rightarrow x = \dfrac{7}{2}andx = \dfrac{{ - 3}}{4}$
So (C) will be the correct option.