Question

How do you solve \[{\left( {2x + 5} \right)^2} = {\left( {2x + 3} \right)^2}\] ?

Answer 1

Hint: The problem is simply related to the algebraic identities. We will square both the sides first. Then we will take the terms having the same coefficient with each other and will perform the necessary calculations. Then we will come across the value of x that satisfies the equation above.

Complete step-by-step answer:
Given that,
\[{\left( {2x + 5} \right)^2} = {\left( {2x + 3} \right)^2}\]
We know that,
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
So applying on the brackets above,
\[{\left( {2x} \right)^2} + 2 \times 2x \times 5 + {5^2} = {\left( {2x} \right)^2} + 2 \times 2x \times 3 + {3^2}\]
Taking the necessary squares and performing the multiplications, also we will cancel the first square term,
\[20x + 25 = 12x + 9\]
Now taking the variable term on one side and constants on other side,
\[20x - 12x = 9 - 25\]
On subtracting we get,
\[8x = - 16\]
On dividing ,
\[x = \dfrac{{ - 16}}{8}\]
\[x = - 2\]
This is our final answer.
So, the correct answer is “\[x = - 2\]”.

Note: Here use of algebraic identities is the notable part. Also if the term of square if was different then we might have got a quadratic equation which we need to solve using quadratic equation method. Also note that the terms with same coefficients can either be added or subtracted.