Question

How do you solve \[22x - {x^2} = 96\]?

Answer 1

Hint: To solve the given equation, we need to do factor out the terms, i.e., 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:
\[22x - {x^2} = 96\]
Let’s, rearrange the terms as:
\[ - {x^2} + 22x = 96\]
Now, move the terms of RHS to LHS as,
\[ - {x^2} + 22x = 96\]
\[ \Rightarrow - {x^2} + 22x - 96\]
Now, we have all numbers to one side of the equation, hence make it equal to 0 i.e.,
\[ \Rightarrow - {x^2} + 22x - 96 = 0\]
Take out the common factor as:
\[ \Rightarrow - 1\left( {{x^2} - 22x + 96} \right) = 0\]
As, the equation is of the form \[a{x^2} + bx + c\], hence to find the factors, we need to use sum-product pattern as:
\[ - 1\left( {{x^2} - 22x + 96} \right) = 0\]
The pair of integers we need to find for product is c and whose sum is b, in which here the product is \[ + 96\]and sum is \[ - 22\].
\[ \Rightarrow - 1\left( {{x^2} - 6x - 16x + 96} \right) = 0\]
Now, take the common factor from the two pairs:
\[ - 1\left( {\left( {{x^2} - 6x} \right) + \left( { - 16x + 96} \right)} \right) = 0\]
Taking x and 16 as a common term we get:
\[ \Rightarrow - 1\left( {x\left( {x - 6} \right) - 16\left( {x - 6} \right)} \right) = 0\]
Rewrite the obtained terms in factored form as:
\[ \Rightarrow - 1\left( {x - 16} \right)\left( {x - 6} \right) = 0\]
Hence, the factors are
\[\left( {x - 16} \right)\left( {x - 6} \right) = 0\]
Now, we need to create separate equations for the obtained factors, hence
\[\left( {x - 16} \right) = 0\]
\[\left( {x - 6} \right) = 0\]
Let us solve for the first factor i.e.,
\[\left( {x - 16} \right) = 0\]
Therefore, rearrange and isolate the variable to find the solution:
\[ \Rightarrow x = 16\]
Now, let us solve for the second factor i.e.,
\[\left( {x - 6} \right) = 0\]
Therefore, we get
\[ \Rightarrow x = 6\]

Hence, the solution is:
\[x = 6\] and \[x = 16\].


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.