Question

How do you factor the trinomial \[7{x^2} - 31x + 12\]?

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 factor by grouping and combine all the like terms, in which we can easily find the factors of the equation using AC method.

Complete step by step solution:
Let us write the given equation
\[7{x^2} - 31x + 12\]
We can consider that the equation is of the form \[a{x^2} + bx + c\], by which we can easily find the factors of the equation using AC method.
Using the AC method of factorising a quadratic, consider the factors of \[\left( {7 \times 12} \right)\] which sum to - 31 the factors of +84 which sum to - 31 are - 3 and - 28 splitting the middle term gives:
\[7{x^2} - 31x + 12\]
Use the sum-product pattern we get:
\[ \Rightarrow 7{x^2} - 3x - 28x + 12\]
Now let us take out the common factor from the two pairs and simplify all terms, we get
\[7{x^2} - 3x - 28x + 12\]
\[ \Rightarrow x\left( {7x - 3} \right) - 4\left( {7x - 3} \right)\]
Hence, the factors are
\[\left( {x - 4} \right)\left( {7x - 3} \right)\]
And to solve the equation:
\[\left( {x - 4} \right)\left( {7x - 3} \right) = 0\]
Equate each of the factors to zero and solve for x i.e.,
\[\left( {x - 4} \right) = 0\]
\[\left( {7x - 3} \right) = 0\]
Now let us solve for the first factor i.e.,
\[x - 4 = 0\]
Therefore, we get
\[x = 4\]
Now let us solve for the second factor i.e.,
\[7x - 3 = 0\]
\[ \Rightarrow 7x = 3\]
Therefore, we get
\[ \Rightarrow x = \dfrac{3}{7}\]
So, the correct answer is “\[ x = \dfrac{3}{7}\] and \[x = 4\] ”.

Note: The key point to factor the trinomial equation is to group the factors 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.