Question

The roots of the equation \[{x^2} - 3x - 10 = 0\] are
A.Roots are 1 and 3
B.Roots are -1 and 5
C.Roots are -8 and 2
D.Roots are 4 and 3

Answer 1

Hint:First get a quadratic expression from the given equation. Now take the 3 coefficients and mark them as a, b, c. Find the value of product ac. Now try to write b as the sum or difference of 2 numbers which when multiplied you must get ac. Now take the common from first two terms and last two terms such that there must be common terms in both the above terms. Basically, we are converting the quadratic into multiplication of 2 unit degree terms.

Complete step-by-step answer:
Factorization: For factoring a quadratic follow the steps:
Allot \[{x^2}\] coefficient as “a”, x coefficient as “b”, constant as “c”.
Find the product of the 2 numbers a,c. Let it be K.
Rewrite the term bx in terms of those 2 numbers.
Now factor the first two terms.
If we do it correctly our 2 new terms will have one more common factor.
Now take that common to get quadratic as a product of 2 terms, from which you get the roots.
Given expression in the question can be written as:
\[{x^2} - 3x - 10 = 0\]
Here, we get \[{\rm{a}} = {\rm{1}},{\rm{b}} = - {\rm{3}},{\rm{c}} = - {\rm{1}}0\] . So, product ac is \[{\rm{ac}} = - {\rm{1}}0\].
The 2 numbers whose sum is -3, products is -10 are 2, -5.
So, we can write the quadratic equation in the form:
\[{x^2} + 2x - 5x - 10 = 0\]
By taking x common from first two terms, we get it as:
\[x\left( {x + 2} \right) - 5x - 10{\rm{ = 10}}\]
By taking -5 common from last two terms, we get it as:
\[x\left( {x + 2} \right) - 5\left( {x + 2} \right) = 0\]
By taking (x+2) common from whole equation, we get it as:
\[x + 2 = 0{\rm{ ; x - 5 = 0}}\]
From these we get the roots as given below:
\[x = - 2\,\,\,\,\,x = 5\]

Note: The idea of getting the 2 numbers fitting the required condition is the only crucial element, do it properly. Instead of \[{\rm{2x}} - {\rm{5x}}\], you can write\[ - {\rm{5x}} + {\rm{2x}}\], you take x as common from first two terms and +2 as common from last two terms. By this you’ll get the same solutions. So, it doesn’t matter in which order you use the 2 numbers.