Question

Given that ${z^2} - 10z + 25 = 9$, what is the value of z?
(A) $3,4$
(B) $1,6$
(C) $2,6$
(D) $2,8$

Answer 1

Hint: Given equation is of degree $2$. Equations of degree $2$ are known as quadratic equations. Quadratic equations can be factored by the help of splitting the middle term method. In this method, the middle term is split into two terms in such a way that the equation remains unchanged and is then factored using the algebraic operations.

Complete step by step answer:
For factorising the given quadratic equation ${z^2} - 10z + 25 = 9$ , we use the splitting the middle term method.
Firstly, we transpose the constant terms from the right side of the equation to the left side of the equation.
So, ${z^2} - 10z + 25 = 9$
$ \Rightarrow {z^2} - 10z + 25 - 9 = 0$
Simplifying the equation, we get,
$ \Rightarrow {z^2} - 10z + 16 = 0$
Now, we have to factorise the quadratic equation thus obtained. We can use the splitting the middle term method in which the middle term is split into two terms such that the sum of the terms gives us the original middle term and the product of the terms gives us the product of the constant term and coefficient of ${x^2}$.
$ \Rightarrow {z^2} - \left( {8 + 2} \right)z + 16 = 0$
We split the middle term $ - 10z$ into two terms $ - 8z$ and $ - 2z$ since the product of these terms, $16{z^2}$ is equal to the product of the constant term and coefficient of ${z^2}$ and sum of these terms gives us the original middle term, $ - 10z$.
$ \Rightarrow {z^2} - 8z - 2z + 16 = 0$
Taking $z$ common from the first two terms and $ - 2$ common from the last two terms. We get,
$ \Rightarrow {z^2} - 8z - 2z + 16 = 0$
$ \Rightarrow z\left( {z - 8} \right) - 2\left( {z - 8} \right) = 0$
Factoring the common expression out of the bracket, we get,
$ \Rightarrow \left( {z - 2} \right)\left( {z - 8} \right) = 0$
Now, since the product of two terms is equal to zero. Since, one of the two terms have to be zero.
So, either $\left( {z - 2} \right) = 0$ or $\left( {z - 8} \right) = 0$.
Simplifying the equation and finding the values of z, we get,
$z = 2$ or $z = 8$
Hence, the values of z are: $2$ and $8$.

So, the correct answer is “Option D”.

Note:
Besides splitting the middle term, there are various methods to solve quadratic equations such as completing the square method and using the Quadratic formula that can be used to solve for the values of z. Splitting the middle term can be a tedious process at times when the product of the constant term and coefficient of ${x^2}$ is a large number with a large number of divisors. Special care should be taken in such cases.