Question

How do you factor: \[y = 2{x^2} + 13x + 15.\]

Answer 1

Hint: Here we have to factor the given expression by using simple calculation. Then we will use middle term factorisation to split the above function into multiplication of two terms. Finally we get the required answer.

Formula used: Middle term factorisation helps to split the middle term into two terms, and then we need to take common terms from those terms.
Suppose, we want to convert \[{a^2} + 2ab + {b^2}\] into multiplication of two factors .
Then we can split up \[2ab\] into \[ab + ab\].
So, we can imply it into following form:
\[{a^2} + 2ab + {b^2}\]
\[ = {a^2} + ab + ab + {b^2}\]
\[ = a(a + b) + b(a + b)\]
\[ = (a + b)(a + b)\].

Complete step-by-step solution:
The given equation is \[y = 2{x^2} + 13x + 15.\]
First of all we will take L.C.M of constant term in \[{x^2}\]and the only constant term left in the equation.
So, we will do L.C.M for \[2\] and \[15\].
So, \[L.C.M(2,15)\] will be as following:
\[2 = 1 \times 2\] and \[15 = 5 \times 3 \times 1\].
As there are no terms common in \[2\] and \[15\], the L.C.M of \[2\] and \[15\] will become\[ = 2 \times 15 = 30.\]
So, we can rewrite the given equation as following:
\[y = 2{x^2} + 10x + 3x + 15\], as \[L.C.M(2,15)\] is \[30\], can be written as \[(3 \times 10)\].
So, after splitting up the middle term of the above quadratic equation, we need to take the common of constant terms to make it as a form of two same terms.
 As it is clearly visible that in first two terms \[2{x^2}\] and \[10x\]can be written as \[(2x \times x)\] and \[(2x \times 5)\], also, \[3x\]and \[15\] can be written as \[(3 \times x)\] and \[(3 \times 5)\].
So, the equation becomes:
\[y = 2x \times x + 2x \times 5 + 3 \times x + 3 \times 5\].
So, it is clearly visible that the first two terms have a common term of \[2x\] and the last two terms have a common term of \[3\].
So, we can rewrite the equation as:
\[y = 2x(x + 5) + 3(x + 5)\].
So, in the above equation, we can directly say that it has a common term of \[(x + 5)\].
So, we can rewrite the equation as:
\[y = (x + 5)(2x + 3)\].

\[\therefore \] The final factorisation of \[2{x^2} + 13x + 15\] will be \[(x + 5)(2x + 3)\].

Note: Points to remember that:
\[1.\] the equation should be in the form of \[y = a{x^2} + bx + c\].
\[2.\] splitting the middle term in such a way that their product is equal to the product of first and last terms of the trinomial