Question

How do you factor the expression ${{d}^{2}}+10d+9$?

Answer 1

Hint: We can factor the given expression ${{d}^{2}}+10d+9$ by the method of factorization. We will do the given question by the method of splitting the constant and doing the sum product pattern. So, guess two numbers having sum as 10 and product as 9.

Complete step by step answer:
From the given question the expression is, ${{d}^{2}}+10d+9$ .
Factorization is the process of finding factors of a number are numbers that divide evenly into another number. Factorization writes a number as the product of smaller numbers. Factorization is the process of reducing the bracket of a reducing the bracket of a quadratic equation, instead of expanding the bracket and converting the equation to a product of factors which cannot be reduced further. There are many methods for the factorization process.
Now, we should split into the product of two numbers so that the sum of the two numbers must be equal to coefficient of $d$
$9=1\times 9$
We can split $9$ into the product of $1\text{ and 9}$ and their sum $10$
Now the given expression can be written as,
${{d}^{2}}+10d+9$
$\Rightarrow {{d}^{2}}+d+9d+9$
$\Rightarrow d\left( d+1 \right)+9\left( d+1 \right)$
$\Rightarrow \left( d+9 \right)\left( d+1 \right)$
Hence the given expression can be factored as $\left( d+9 \right)\left( d+1 \right)$
By zero theorem,
$\left( d+9 \right)\left( d+1 \right)=0$
$d=-9\text{ and d=-1}$ are the factors for the given expression.

Note: We should be well known about the process of the factorization. We should be careful while splitting the constant into the product of the two numbers. We should be careful that the two numbers sum must be equal to the coefficient of the variable of degree one. This can also be done using the method to find zeros of any quadratic expression $a{{x}^{2}}+bx+c$ given by the formulae $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . For this question ${{d}^{2}}+10d+9$ it will be given as $\dfrac{-10\pm \sqrt{100-4\left( 9 \right)}}{2}=\dfrac{-10\pm \sqrt{64}}{2}=\dfrac{-10\pm 8}{2}=-9,-1$ are factors.