Question

How do you solve the equation ${x^2} + 4x = 21$?

Answer 1

Hint: Factorizing reduces the higher degree equation into its linear equation. In the above given question, we need to reduce the quadratic equation into its simplest form in such a way that addition of products of the factors of first and last term should be equal to the middle term i.e. $4x$.

Complete step by step answer:
The equation $a{x^2} + bx + c$ is a general way of writing quadratic equations where a, b and c are numbers.
In the above expression, can be written as
$ \Rightarrow {x^2} + 4x - 21 = 0$
a=1, b=4, c=-21
First step is by multiplying the coefficient of ${x^2}$ and the constant term -21, we get $ - 21{x^2}$.
After this, factors of $ - 21{x^2}$should be calculated in such a way that their addition should be equal to 4x.
Factors of 21 can be 7 and -3.
Since, we consider negative sign of 21 one of the factors should be negative so that multiplication result is negative and by adding these two factors we should get positive \[14x\].
Therefore, we can add in the following way $7{x^{}}$+ $( - 3x)$=$4x$.
So, further we write the equation by equating it with zero and splitting the middle term according to the factors.
$
   \Rightarrow {x^2} + 4x - 21 = 0 \\
   \Rightarrow {x^2} + 7x - 3x - 21 = 0 \\
 $
Now, by grouping the first two and last two terms we get common factors.
\[
   \Rightarrow x(x + 7) - 3(x + 7) = 0 \\
   \Rightarrow (x - 3)(x + 7) = 0 \\
 \]
Taking x common from the first group and -3 common from the second we get the above equation.
Therefore, by solving the above quadratic equation we get factors 3 and - 7.

Note: In quadratic equation, an alternative way of finding the factors is by using a formula which is given below:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
By substituting the values of a=1, b=4 and c=-21 we get the factors of x.
$
\Rightarrow x = \dfrac{{ - (4) \pm \sqrt {{{(4)}^2} - 4(1)( - 21)} }}{{2(1)}} \\
 $
So, the values are \[x = 3\] or \[x = - 7\].