Question

Find the factors of the expression, ${x^2} - 10x + 24$.

Answer 1

Hint:Check if the given expression is arranged in the same form as the standard
expression,$a{x^2} + bx + c$. Find the factors of the number got by multiplying “a” and “c”. Arrange
the factors in such a way that it is their sum is equal to the coefficient of $x$. Once the expression is
factorised, equate the resulting expression with zero to find the factors.

Complete step by step solution:
Comparing the given expression to the standard equation, $a{x^2} + bx + c$

We see that, $a = 1,b = - 10\;\& \;c = 24$.

Now, we need to find the factors of $a \times c = 1 \times 24 = 24$. They will be 1,2,3,4, 6, 12 and
24.

We can clearly see that, -6-4 = -10, which is equal to the coefficient of x.

Thus, we can now write \[{x^2} - 10x + 24 = {x^2} - 6x - 4x + 24\,\],
=\[x(x - 6) - 4(x - 6)\]\[.......\;(Taking\;x\,and\;4\;common)\]
=\[(x - 6)(x - 4).....(Taking\;x - 6\;common)\]

Now that we have factored the expression, we will equate the expression with 0 to find the factors.
\[
{x^2} - 10x + 24 = 0 \\
\Rightarrow (x - 6)(x - 4) = 0 \\
\]

This means we can equate both the above terms to 0.
$
(x - 6) = 0,\,(x - 4) = 0 \\
x = 6,x = 4 \\
$

Thus, the expression ${x^2} - 10x + 24$has 2 factors, 6 and 4.

Note: This method of factorization is called the “mid-term” factorization. It can be used only if the coefficient of x can be written as the sum of any two factors of the product of the constant and the coefficient of ${x^2}$ . It becomes easier to use this method only if one is dealing with integers. In case of fractional or imaginary values, this method is very cumbersome and hence should not be used. In such a case, you can use the formula, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$to find the factors. \[\]