Find algebraically the solution set of the equation the absolute value of 𝑥 squared plus 13𝑥 plus 21 is equal to 21.
In this question, we’re asked to determine the solution to an absolute value equation algebraically. This means although we could attempt to solve this by using a graphical approach, we need to use an algebraic approach. To do this, we first recall when we take the absolute value of a number, we want to take the size of the number. In other words, we don’t mind its sign. It can be negative or positive. And this is useful for helping us solve this equation. If the absolute value of some number is equal to 21, then that number must be 21 or negative 21. These are the only two numbers whose absolute value is 21. Therefore, 𝑥 squared plus 13𝑥 plus 21 is either equal to 21 or negative 21. So we could find all of the values of 𝑥 which solve this equation by solving both of these two quadratic equations.
Let’s start with the quadratic equation on the left. We can solve this by subtracting 21 from both sides. This gives us that 𝑥 squared plus 13𝑥 will be equal to zero, and we can solve this by factoring. We’ll take out the shared factor of 𝑥. This gives us that 𝑥 times 𝑥 plus 13 is equal to zero. And for the product of two numbers to be equal to zero, one of the factors must be zero. Therefore, we get two solutions: either 𝑥 is zero or 𝑥 is negative 13. And it’s worth noting we could substitute these values into our original equation to check that they are indeed solutions.
We can do the same to solve the second equation. We’ll start by adding 21 to both sides. This gives us that 𝑥 squared plus 13𝑥 plus 42 is equal to zero. We can try and solve this equation by factoring. We need two numbers which multiply to give 42 and add to give 13. And by considering the factors of 42, we could notice that six multiplied by seven is 42 and six plus seven is 13. So this factors to give us 𝑥 plus six multiplied by 𝑥 plus seven is equal to zero. And once again for the product of numbers to be equal to zero, one of the factors must be zero. In this case, either 𝑥 is negative six or 𝑥 is negative seven.
And just as we did before, we could substitute these values back into our original equation to check that they are indeed solutions. For example, if we substitute 𝑥 is equal to negative six into this equation, we get the absolute value of 36 minus 78 plus 21 should be equal to 21. And the left-hand side of this equation simplifies to be the absolute value of negative 21. So this is a solution to the equation. Finally, the question wants us to write this as a solution set. That’s the set of all solutions to the equation. Doing this, we get our final answer. The solution set to the equation the absolute value of 𝑥 squared plus 13𝑥 plus 21 is equal to 21 is the set containing negative 13, negative seven, negative six, and zero.