Question

How do you solve ${x^2} + {(x - 3)^2} = 225$?

Answer 1

Hint: In the question, we have an equation that is quadratic, and we have to find the value of $x$, which can be found with the help of finding its factors, which can be done with the help of rules. There are many rules and formulas that help in factorising equations; we just have to observe which condition the equation satisfies.
Here, we can apply the formula ${(a - b)^2} = {a^2} - 2ab + {b^2}$ and then factorise it further, find its roots and know the value of the variable.

Complete step-by-step solution:
For an equation ${x^2} + {(x - 3)^2} = 225$, we can observe the bracket and we have a formula to open the bracket by squaring principle,
$ \Rightarrow {x^2} + {(x - 3)^2} = 225$
Please note that,
\[ \Rightarrow {(x - y)^2} = {x^2} - 2xy + {y^2}\]
Therefore, for the bracket with the square,
\[ \Rightarrow {(x - 3)^2} = {x^2} - 6x + 9\]
Placing the values in the equation to solve it further,
\[ \Rightarrow {x^2} + ({x^2} - 6x + 9) = 225\]
Opening the brackets,
$ \Rightarrow {x^2} + {x^2} - 6x + 9 = 225$
Adding like variables and shifting constants,
\[ \Rightarrow 2{x^2} - 6x + 9 - 225 = 0\]
Therefore, the value of equation will be,
$\, \Rightarrow 2{x^2} - 6x - 216 = 0\;$
If we simplify, we obtain the equation below,
\[ \Rightarrow {x^2} - 3x - 108 = 0\]
Now, let us factor the equation above,
\[ \Rightarrow (x + 9)(x - 12) = 0\]
If \[(x + 9) = 0\;\] then \[x = - 9\]
If \[(x - 12) = 0\;\;\] then \[x = 12\]

Therefore, for an equation ${x^2} + {(x - 3)^2} = 225$ the values will be if \[(x + 9) = 0\;\] then \[x = - 9\]
If \[(x - 12) = 0\;\;\] then \[x = 12\]

Note: In factorisation, one must be aware of the fact that the roots of an equation can change and also, the solution, if there exist two roots for an equation. For example, in the above equation we have two roots for the equation and their solution changes as per the roots changed. Factorisation is the reverse of multiplying out. One important factorisation process is the reverse of multiplications. The factors of any equation can be an integer, a variable or an algebraic expression itself.