Question

How to solve \[{b^2} + 5b - 35 = 3b\] by factoring?

Answer 1

Hint:To solve the given equation by factoring, combine all the like terms we get the form of \[{x^2} + bx + c\], by which we can easily find the factors of the equation using AC method.

Complete step by step answer:
Let us write the given equation
\[{b^2} + 5b - 35 = 3b\]
Shift the term \[3b\] from RHS to the LHS and the equation we get is
\[{b^2} + 5b - 35 - 3b = 0\]
Now let us combine and simplify all the like terms, we get
\[{b^2} + 2b - 35 = 0\]
We can consider that the obtained equation is of the form \[{x^2} + bx + c\], by which we can easily find the factors of the equation using the AC method.
\[{b^2} + 2b - 35 = 0\]
The pair of integers we need to find for the product is c and whose sum is b, in which the product is -35 and sum is 2.
\[{b^2} + 7b - 5b - 35 = 0\]
Hence, the factors are
\[\left( {b + 7} \right)\left( {b - 5} \right) = 0\]
If any individual factor of the equation is equal to zero, then this implies that the entire expression is equal to zero.
\[\left( {b + 7} \right) = 0\]
\[\left( {b - 5} \right) = 0\]
Now let us solve for the first factor i.e.,
\[\left( {b + 7} \right) = 0\]
Therefore, we get
\[b = - 7\]
Now let us solve for the second factor i.e.,
\[\left( {b - 5} \right) = 0\]
Therefore, we get
\[b = 5\]
Hence, the final solution is true for
\[b = 5, - 7\]

Note: The key point to find the equation using factoring method i.e., of the form \[{x^2} + bx + c\], in this given quadratic equation we need to find two integers whose product is equal to c and the sum is equal to b using AC method. Then solve each factor obtained by setting it to zero by this we can get the value of b of both the factors.