Question

Solve the given equation for the variable “x”
\[x(2x + 5) = 25\]?

Answer 1

Hint: Here the given question is to solve for the variable by simplifying the equation, Here to simplify we have to solve the bracket then a quadratic equation will form, in order to solve that equation either sridharacharya rule or mid term splitting rule will be considered.

Formulae used: Discriminant for the quadratic equation:
\[ \Rightarrow d = \sqrt {{b^2} - 4ac} \]
Roots of the quadratic equation:
\[ \Rightarrow roots = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step-by-step answer:
The given question is \[x(2x + 5) = 25\]
On simplifying we get:
\[
   \Rightarrow x(2x + 5) = 25 \\
   \Rightarrow 2{x^2} + 5x - 25 = 0 \\
 \]
Here mid term splitting rule cannot be used, since middle term cannot be broken according to the rule, so here we will use sridharacharya rule:
\[ \Rightarrow d = \sqrt {{b^2} - 4ac} = \sqrt {{5^2} - 4 \times 2 \times ( - 25)} = \sqrt {225} = 15\]
Here we get the discriminant “d” as 15
Now roots of equations are:
\[
   \Rightarrow roots = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - 5 \pm 15}}{{2 \times 2}} \\
   \Rightarrow roots = \dfrac{{ - 5 + 15}}{4},\dfrac{{ - 5 - 15}}{4} \\
   \Rightarrow roots = 2, - 5 \\
 \]
Here we got the final solution of the equations.

Note: Here we got the quadratic equation after simplifying the equation and solving accordingly. To solve the quadratic equation we have to either use the sridharacharya rule or use the mid term splitting rule to solve for the roots of the equation.