Question

How do u factor \[5{x^2} - 31x + 6\]?

Answer 1

Hint: To solve the given equation by factoring, find two numbers that multiply to give a times c and add to give b of the quadratic equation of the form \[a{x^2} + bx + c\], in which we need to combine all the like terms we get the form of \[a{x^2} + bx + c\], 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:
 \[5{x^2} - 31x + 6\]
The first term is, \[5{x^2}\] and its coefficient is 5.
The middle term is, \[ - 31x\] and its coefficient is -31.
The last term, "the constant", is +6.
We can consider that the obtained equation is of the form \[a{x^2} + bx + c\], by which we can easily find the factors of the equation using the AC method.
Here we need to find two factors of 30 whose sum equals the coefficient of the middle term, which is -31, hence rewrite the polynomial splitting the middle term using the two factors i.e., -30 and -1, we have:
 \[5{x^2} - 31x + 6\]
 \[ \Rightarrow 5{x^2} - 30x - 1x + 6\]
Add up the first 2 terms, pulling out like factors:
 \[ \Rightarrow 5x \cdot \left( {x - 6} \right)\]
 Add up the last 2 terms, pulling out common factors:
 \[ \Rightarrow 1 \cdot \left( {x - 6} \right)\]
Add up the four terms we get:
 \[ \Rightarrow \left( {5x - 1} \right) \cdot \left( {x - 6} \right)\]
The pair of integers we need to find for the product is c and whose sum is b, in which here the product is 6 and sum is -31.
Hence, the factors are
 \[\left( {5x - 1} \right)\left( {x - 6} \right)\]
When a product of two or more terms equals zero, then at least one of the terms must be zero. We shall now solve each term = 0 separately as:
 \[\left( {5x - 1} \right) = 0\] ……………… 1
 \[\left( {x - 6} \right) = 0\] ……………… 2
So, let us solve for the first factor i.e., equation 1:
 \[5x - 1 = 0\]
Therefore, we get
 \[ \Rightarrow 5x = 1\]
 \[ \Rightarrow x = \dfrac{1}{5} = 0.20\]
Now let us solve for the second factor i.e., equation 2:
 \[x - 6 = 0\]
 \[ \Rightarrow x = 6\]
Therefore, the points are:
 \[x = \dfrac{1}{5} = 0.20\] and \[x = 6\].
So, the correct answer is “ \[x = \dfrac{1}{5} = 0.20\] and \[x = 6\]”.

Note: The key point to find the equation using factoring method i.e., of the form \[a{x^2} + bx + c\], 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.