Question

Find the solution of \[{n^2} - 2n - 35\].

Answer 1

Hint: A perfect square trinomial on the left side of the equation should be created. Then it should be factored and hence solve for “x”. The general equation for a perfect square trinomial is\[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\]. Completing the square is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial.

Complete step by step answer:
To solve quadratics by factoring, we use something called "the Zero-Product Property". If we multiply two (or more) things together and the result is equal to zero, then we know that at least one of those things that we multiplied must also have been equal to zero. Putting it another way, the only way for us to get zero when we multiply two (or more) factors together is for one of the factors to have been zero.

So, if we multiply two (or more) factors and get a zero result, and then we know that at least one of the factors was itself equal to zero. In particular, we can set each of the factors equal to zero, and solve the resulting equation for one solution of the original equation.Solving the above equation we have,
\[{n^2} - 2n - 35 \\
\Rightarrow {n^2} - 7n + 5n - 35 \\
\Rightarrow n\left( {n - 7} \right) + 5\left( {n - 7} \right) \\
\therefore\left( {n - 7} \right)\left( {n + 5} \right) \\
 \]
Hence the factors are \[\left( {n - 7} \right)\left( {n + 5} \right)\].

Note:To solve an equation by completing the square method the constant term should be alone on the right side. If the leading coefficient “a” that is the coefficient of \[{x^2}\] terms not equal to 1 then both sides should be divided by “a”. The square of half the coefficient of the x-term should be added to both sides of the equation and the left side should be factored as the square of a binomial. Both the sides should be taken square root and then x should be solved.