Question

How do you write ${{d}^{2}}-18d+80$ in factored form?

Answer 1

Hint: We will factor the given quadratic equation by using the splitting the middle term method. We will split the middle term of the equation $a{{x}^{2}}+bx+c=0$ such that the product of two numbers is equal to $a\times c$ and the sum of those two numbers is equal to $b$.

Complete step-by-step solution:
We have been given an equation ${{d}^{2}}-18d+80$.
We have to write the given equation in factored form.
Now, we will use the split middle term method. We have to find two numbers such as the product of two numbers is equal to $a\times c=1\times 80=80$ and their sum is equal to $b=18$.
So we will use two numbers as 10 and 8.
So splitting the middle term we will get
$\Rightarrow {{d}^{2}}-\left( 10d+8d \right)+80$
Now, simplifying the above obtained equation we will get
$\Rightarrow {{d}^{2}}-10d-8d+80$
Now, taking the common terms out we will get
$\Rightarrow d\left( d-10 \right)-8\left( d-10 \right)$
Now, again taking common factors out we will get
$\Rightarrow \left( d-10 \right)\left( d-8 \right)$
Hence we get the factors of the given equation as $\left( d-10 \right)\left( d-8 \right)$.

Note: Here in this question we use the split middle term method as it is a simple question. We can also use other methods like quadratic formula, completing the square method also to solve the quadratic equations. Also we can find the values of x by equating each factor to zero and by solving the obtained equations.