Find the value of 𝑐 given that 𝑛 of 𝑥 is equal to 14 divided by 25𝑥 squared plus 60 𝑥 plus 36, where 𝑛 of 𝑐 is undefined.
A function is said to be undefined if the denominator equals zero. Therefore, to find a value, or values, we set the denominator, in this case 25𝑥 squared plus 60 𝑥 plus 36, equal to zero.
We can solve this quadratic equation by factorising, by completing the square, or by using the quadratic formula. We will firstly look at solving it by factorising. As there is no common factor with the exception of one, we are going to factorise this into two brackets or parentheses.
Five 𝑥 multiplied by five 𝑥 is 25𝑥 squared. And six multiplied by six is equal to 36. As all the terms in the quadratic equation were positive, the two signs in our parentheses need to be positive signs or addition signs. We can check that this factorisation is correct by using our FOIL method.
Multiplying the first terms gives us 25𝑥 squared. Multiplying the outside terms, five 𝑥 and positive six, gives us 30𝑥. Multiplying the inside terms also gives us 30𝑥. And finally, multiplying the last terms, six multiplied by six, gives us 36.
Grouping the like terms to simplify this equation gives us 25𝑥 squared plus 60𝑥 plus 36. Therefore, our factorisation was correct. You will notice that when we factorised our equation, both of the brackets, or parentheses, were exactly the same. This means that instead of getting two solutions, as we normally would when solving a quadratic equation, we will only get one solution or value.
In order to calculate this value, we will solve the equation five 𝑥 plus six equals zero. Subtracting six from both sides of the equation leaves us with five 𝑥 equals negative six. And finally dividing both sides by five gives us a final answer of 𝑥 equal to negative six over five or minus six-fifths. Therefore, the value of 𝑐 where the function 𝑛 of 𝑐 is undefined is negative six-fifths.