Question

The square of a positive number is $21$ more than $4$ times the number. How do you find the number?

Answer 1

Hint: First of all convert the given word statements in the form of mathematical expression and then find the factors of the term and then get the positive term out of it. First of all assume any reference number as “x”.

Complete step by step solution:
Let us assume the number be “x”
Now, convert the word statements in the form of mathematical expression –
square of a positive number is $21$ more than $4$ times the number
$ \Rightarrow {x^2} = 21 + 4x$
Now to get the value of “x” frame the above equation to find the factors.
$ \Rightarrow {x^2} - 4x - 21 = 0$
Now, to get the factors we will use the concept of to split the middle term.
Here we have three terms in the given expression.
Now, multiply the constant in the first term with the last term.
i.e. $1 \times ( - 21) = ( - 21)$
Now, you have to split the middle term to get $ - 21$ in multiplication and addition or subtraction to get the middle term i.e. $( - 4)$ . Here applying the basic concept of the product of two negative terms gives us the positive term and addition of two negative terms gives the value in the negative sign.
$
 \Rightarrow ( - 21) = ( - 7) \times (3) \\
 \Rightarrow ( - 4) = ( - 7) + 3 \\
 $
Write the equivalent value for the middle term –
$ \Rightarrow {x^2} - 7x + 3x - 21 = 0$
Now, make the pair of two terms in the above equation-
$ \Rightarrow \underline {{x^2} - 7x} + \underline {3x - 21} = 0$
Find the common factors from the paired terms –
$ \Rightarrow x(x - 7) + 3(x - 7) = 0$
Take the common factors in the above equation –
$ \Rightarrow (x - 7)(x + 3) = 0$
$ \Rightarrow x = 7, - 3$

Hence, the required positive term is $7$

Note: Here we were able to split the middle term and find the factors but in case it is not possible then we can find factors by using the formula \[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\] and considering the general form of the quadratic equation $a{x^2} + bx + c = 0$ . Be careful about the sign convention and simplification of the terms in the equation.