Question

Solve the equation ${\left( {x + 4} \right)^2} - {\left( {x - 5} \right)^2} = 9$

Answer 1

Hint: In the given question, we have to simplify the expression and solve for the value of x. We first compute the whole squares of the terms using algebraic identities ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ and ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ given in the equation and then cancel the like terms having opposite signs. Then, we use the method of transposition to isolate the variable x and find its value.

Complete step-by-step solution:
So, we have the equation ${\left( {x + 4} \right)^2} - {\left( {x - 5} \right)^2} = 9$.
So, we first evaluate the terms involving whole squares using the algebraic identities ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ and ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. So, we get,
$ \Rightarrow \left( {{x^2} + 2\left( x \right)\left( 4 \right) + {4^2}} \right) - \left( {{x^2} - 2\left( x \right)\left( 5 \right) + {5^2}} \right) = 9$
Computing the squares of the numbers and simplifying the calculations, we get,
$ \Rightarrow \left( {{x^2} + 8x + 16} \right) - \left( {{x^2} - 10x + 25} \right) = 9$
Opening the brackets, we get,
$ \Rightarrow {x^2} + 8x + 16 - {x^2} + 10x - 25 = 9$
Cancelling the like terms with opposite signs, we get,
$ \Rightarrow 8x + 16 + 10x - 25 = 9$
Adding up like terms and simplifying calculations,
$ \Rightarrow 18x - 9 = 9$
Shifting all the constant terms to right side of equation, we get,
$ \Rightarrow 18x = 9 + 9$
$ \Rightarrow 18x = 18$
Dividing both sides of equation by $18$, we get,
$ \Rightarrow x = 1$
So, the value of x is $1$.

Note: Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. Transposition method involves doing the same mathematical operation on both sides of the equation. When we do the same mathematical operations on both sides of the equation, the equation remains unchanged. One should remember the squares of numbers $4$ and $5$ to get to the required answer. We can also verify the answer by putting in the value of x in the equation back and getting the left side of the equation equal to the right side of the equation.
So, putting $x = 1$ back in the equation, we get,
${\left( {1 + 4} \right)^2} - {\left( {1 - 5} \right)^2} = 9$
Simplifying the calculations and computing the squares, we get,
$ \Rightarrow {5^2} - {4^2} = 9$
$ \Rightarrow 25 - 16 = 9$
$ \Rightarrow 9 = 9$
Now, the left side of the equation is equal to the right side. We have verified that $x = 1$ is the solution of the equation given to us.