Question

How do you factor the expression ${{x}^{2}}-11x+24=0$ ?

Answer 1

Hint: As, quadratic equation are those equation in which maximum power of variable is $2$ it is in form of $a{{x}^{2}}+bx+c=0$
Where, $a$ is coefficient of $x$
$b$ is coefficient of $x$
And $c$ is constant.
Here, value of $a=1,b=-11,c=24$
For factorising, we need two numbers which when added or subtracted together to form the product of $'a'$ and $'c'$
Like,
$\Rightarrow$ ${{x}^{2}}+2x-2=0$
$a=1,b=2,c=2$
So, $ac=\left( 1\times 2 \right)=2$
So, factors or number will be $2$ and $1$
Hence, we can write is as,
$\Rightarrow$ ${{x}^{2}}+x-2x-2=0$
Apply this concept to factorise the given expression.

Complete step-by-step answer:
As per question,
We have, quadratic equation,
$\Rightarrow$ ${{x}^{2}}-11x+24=0$
Here, comparing the given equation with general form of quadratic equation i.e. $a{{x}^{2}}+bx+c=0$
We will get,
$a=1,b=-1,c=24$
As, here value of product of $'a'$ and $'c'$ will be,
$\Rightarrow$ $\left( 1\times 24 \right)=24$
So, we need two such number which when multiplied together result to $24$ and when added or subtract result to value of $b$ i.e. $-11.$
Hence,
Factorising the number $24$
We will get,
Number as,
$24=1\times 24$ and $-1\times -24$
$24=2\times 12$ and $-4\times -6$
$24=3\times 8$ and $-3\times -3$
Out of all given factorisation,
Only $-3$ and $-8$ are such numbers which when added result to the value of $b.$ i.e. $-11.$
So,
$\Rightarrow$ ${{x}^{2}}-11x+24=0$
Can be written as,
$\Rightarrow$ ${{x}^{2}}-3x-8x+24=0$
$\Rightarrow$ $x\left( x-3 \right)-8\left( x-3 \right)=0$
$\Rightarrow$ $\left( x-5 \right)\left( x-3 \right)=0$

Hence, factorisation of ${{x}^{2}}-11x+24$ will be $\left( x-8 \right)\left( x-3 \right)$

Additional Information:
Let if we have to factorise
$\Rightarrow$ ${{x}^{2}}+3x+2=0$
Here, signs of all co-efficient are +ve.
So,
$\Rightarrow$ $ac=2\times 1=2$
Hence, two number can be $\left( 2,1 \right)$ and $\left( -1,-2 \right)$
But due to positive sign of $''b''$ we will consider number as $\left( 2,1 \right)$
As,
$+\times +=+$
$+\times -=-$
$-\times +=-$
$-\times -=+$

Note:
In the given equation:
$\Rightarrow$${{x}^{2}}-11x+24=0$
We will not consider $\left( 8,3 \right)$ as pairs of numbers, as here $''b''$ is negative and $''c''$ is positive, so we need to negative numbers, due to that we will consider $\left( -3 \right)$ and $\left( -8 \right)$