Question

Solve the equation, \[{y^2} + 6y - 27 = 0\]

Answer 1

Hint: Here we are asked to solve the given equation. We know that the degree of the equation tells us the number of roots that equation has, here the given equation has a degree two thus this equation will have two roots. First, we will try to break the term containing the variable $y$ then we will rearrange them as factors. After that, we will solve them for the value of $y$ .

Complete answer:
We are given with the equation: \[{y^2} + 6y - 27 = 0\] we aim to solve this equation.
Now we will use the middle term factorization to solve the quadratic equation,
Now we see that \[27\] is the multiplication of \[3,9\] and the negative sign before \[27\] denotes that the middle terms have to be in subtracted form, so:
\[{y^2} + 6y - 27 = 0\]
\[\Rightarrow {y^2} + 9y - 3y - 27 = 0\]
As we can see that the variable $y$ is common in the first two terms and three is common in last two terms so now let us take the term $y$ common from the first two terms and the number three from the last two terms we get,
\[ \Rightarrow y(y + 9) - 3(y + 9) = 0\]
Now we can see that the term \[(y + 9)\] is common in both the terms so let us take the term \[(y + 9)\] commonly out we get,
\[ \Rightarrow (y + 9)(y - 3) = 0\]
From this, we get
$y + 9 = 0,y - 3 = 0$
On simplifying the above, we get
$y = - 9,y = 3$
Therefore, the solutions to the equation are \[y = 3, - 9\]

Note:
It is important that we know various ways, techniques, shortcuts and methods to solve different types of quadratic equations. Middle term factoring, Sridharacharya method or Quadratic formula i.e. \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] or graphical methods are some techniques of solving the roots of a quadratic equation.