Question

How do u factor \[{x^2} + 29x - 30\]?

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
\[{x^2} + 29x - 30\]
The given 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.
Using the AC method of factorising, consider the factors of \[\left( {1 \times \left( { - 30} \right)} \right)\] which sum to 29 the factors of \[ - 30\]which sum to 29 are \[ - 1\] and \[ + 30\]splitting the middle term gives:
\[{x^2} - x + 30x - 30\] i.e., factor by grouping
\[ = x\left( {x - 1} \right) + 30\left( {x - 1} \right)\]
Now let us take out the common factor and simplify all terms, we get the factors as:
\[\left( {x + 30} \right)\left( {x - 1} \right)\]
Hence, the factors are:
\[\left( {x + 30} \right)\left( {x - 1} \right)\]
And now solve the equation:
\[\left( {x + 30} \right)\left( {x - 1} \right) = 0\]
Equate each of the factors to zero and solve for x i.e.,
\[\left( {x + 30} \right) = 0\]
\[\left( {x - 1} \right) = 0\]
Now let us solve for the first factor i.e.,
\[\left( {x + 30} \right) = 0\]
Therefore, we get
\[x = - 30\]
Now let us solve for the second factor i.e.,
\[\left( {x - 1} \right) = 0\]
Therefore, we get
\[x = 1\]
Hence, the values of x are: \[\left( { - 30,1} \right)\].

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.