Question

How do you solve the quadratic equation by completing the square: ${x^2} + 4x = 21$?

Answer 1

First bring the 21 from the RHS to the LHS. Then, write it in such a format that we have ${(x - 2)^2}$ in it and then just simplify the calculations by taking the square – root.

Complete step-by-step solution:
We are given that we are required to solve ${x^2} + 4x = 21$ using the method of completing the square.
Taking 21 from addition in the right hand side to subtraction in the left hand side of the above mentioned equation, we will then obtain the following equation with us:-
$\Rightarrow {x^2} + 4x - 21 = 0$
We can write the above mentioned equation as follows:-
$\Rightarrow {\left( x \right)^2} + 2 \times 2 \times x + {2^2} - {2^2} - 21 = 0$ ……………..(1)
Since we know that we have an identity given by the following formula with us:-
$\Rightarrow {(a + b)^2} = {a^2} + {b^2} + 2ab$
Replacing a by x and b by 2, we will then obtain the following equation with us:-
$\Rightarrow {(x + 2)^2} = {x^2} + {2^2} + 2 \times 2 \times x$
Putting this in equation number 1, we will then obtain the following equation with us:-
$\Rightarrow {\left( {x + 2} \right)^2} - {2^2} - 21 = 0$
Simplifying the left hand side of the above equation, we will then obtain the following equation with us:-
$\Rightarrow {\left( {x + 2} \right)^2} - 4 - 21 = 0$
Simplifying the left hand side of the above equation further, we will then obtain the following equation with us:-
$\Rightarrow {\left( {x + 2} \right)^2} - 25 = 0$
Taking 25 from subtraction in the left hand side to addition in the right hand side, we will then obtain the following equation with us:-
$\Rightarrow {\left( {x + 2} \right)^2} = 25$
Taking square – root of both the sides and taking 2 from addition to subtraction, we will then obtain the following equation with us:-
$\Rightarrow x = - 2 \pm 5$
Thus, we have the required roots as 3 and – 7.

Note: The students must notice that we have an alternate way of factoring the quadratic equation involved in it as well. The alternate way is as follows:-
The given equation is ${x^2} + 4x = 21$.
We can write it as follows:-
$\Rightarrow {x^2} + 4x - 21 = 0$
Using the quadratic formula given by if the equation is given by $a{x^2} + bx + c = 0$, its roots are given by the following equation:-
$\Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Thus, we have the roots of ${x^2} + 4x = 21$ given by:
$\Rightarrow x = \dfrac{{ - 4 \pm \sqrt {{{(4)}^2} - 4 \times ( - 21)} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side, we will then obtain the following equation with us:-
$\Rightarrow x = \dfrac{{ - 4 \pm \sqrt {16 + 84} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side further, we will then obtain the following equation with us:-
$\Rightarrow x = \dfrac{{ - 4 \pm 10}}{2}$
Hence, the roots are 3 and - 7.