Video Transcript
Point 𝐾 lies between points 𝐽 and 𝐿. If 𝐽𝐾 equals 𝑥 squared plus eight 𝑥, 𝐾𝐿 equals three 𝑥 minus two, and 𝐽𝐿 equals 40, find the length of the line segment 𝐽𝐾.
In this question, we have three points, points 𝐾, 𝐽, and 𝐿, which all lie on a straight line. We’re given expressions for the lengths of various segments of this straight line in terms of an unknown variable 𝑥. Firstly, 𝐽𝐾 is 𝑥 squared plus eight 𝑥. So we can label this length on our diagram. Secondly, 𝐾𝐿 is three 𝑥 minus two. So we’ll label this too. Finally, we’re told that 𝐽𝐿, that’s the length of the entire line segment, is 40. We’re asked to find the length of the line segment 𝐽𝐾. And in order to do this, we need to know the value of this unknown variable 𝑥. We can form an equation using the information we’re given.
As the three points lie on a straight line, we know that the length of 𝐽𝐾 plus the length of 𝐾𝐿 is equal to the full length 𝐽𝐿. We can then substitute the expressions we’ve been given. 𝐽𝐾 is 𝑥 squared plus eight 𝑥. 𝐾𝐿 is three 𝑥 minus two. And for 𝐽𝐿, we were given the value 40. So we have an equation, 𝑥 squared plus eight 𝑥 plus three 𝑥 minus two equals 40. To simplify this equation, we first group the like terms on the left-hand side, giving 𝑥 squared plus 11𝑥 minus two is equal to 40.
And then we can subtract 40 from each side so that all of the terms are grouped on the left-hand side. And we have 𝑥 squared plus 11𝑥 minus 42 is equal to zero. This is a quadratic equation in our unknown variable 𝑥. And we need to solve it. There are a number of different methods we could use. We could try factoring the equation, applying the quadratic formula, or completing the square.
If the quadratic equation can be solved by factoring, then this is usually the most efficient way. So let’s try this first. We’re looking for the product of two linear factors which multiply together to give the quadratic 𝑥 squared plus 11𝑥 minus 42. As the coefficient of 𝑥 squared is one, we know that the first term in each set of parentheses must be 𝑥 because 𝑥 multiplied by 𝑥 gives 𝑥 squared.
To complete each set of parentheses, we’re then looking for two values which have a specific set of properties. Firstly, their sum must be equal to the coefficient of 𝑥. So the sum of these two values must be positive 11. Secondly, the product of these two values must be equal to the constant term, which is negative 42.
To help us find these two numbers, we can begin by listing the factor pairs of 42. They are one and 42, two and 21, three and 14, and six and seven. In order for the product to be negative 42 though, we need the signs of the two numbers to be different. So one must be positive and the other negative. If we take our third factor pair and we make the three negative, then we have the numbers negative three and 14, which do have a product of negative 42. And they also have a sum of positive 11. So these are the two numbers we’re looking for to complete our parentheses.
This quadratic factors then as 𝑥 plus 14 multiplied by 𝑥 minus three. And this is equal to zero. We can, of course, check that we factored correctly by redistributing the parentheses, perhaps using the FOIL method, and confirm that this does indeed give the quadratic 𝑥 squared plus 11𝑥 minus 42.
Next, we recall that in order for a product to be equal to zero, at least one of the factors must themselves be equal to zero. So we have either 𝑥 plus 14 equals zero or 𝑥 minus three equals zero. These are straightforward linear equations that can each be solved in one step. To solve the first equation, we subtract 14 from each side, giving 𝑥 equals negative 14. And to solve the second, we add three to each side, giving 𝑥 equals three. So we found two solutions to this quadratic equation: 𝑥 equals negative 14 and 𝑥 equals three. But we need to check whether they’re both suitable values for 𝑥 in the context of this question.
If we look back at our diagram, we see that the length of 𝐾𝐿 was specified as three 𝑥 minus two. If we were to use the value 𝑥 equals negative 14, then the length of this line segment would be three multiplied by negative 14 minus two, which is negative 44. And as this is a negative value, we know that 𝑥 equals negative 14 can’t be the value we’re looking for because the length of a line segment should always be positive. So whilst 𝑥 equals negative 14 is a valid solution to this quadratic equation, it isn’t the value we’re looking for in this problem. The value we’re going to carry forward is 𝑥 equals three.
Now the question didn’t ask us just to find the value of 𝑥; it asked us to find the length of the line segment 𝐽𝐾. So our final step is we need to substitute this value of 𝑥 into the expression for the length of 𝐽𝐾. The expression is 𝑥 squared plus eight 𝑥. So substituting 𝑥 equals three gives three squared plus eight multiplied by three. That’s nine plus 24, which is equal to 33.
So we think we have our answer, but let’s check it. We can do this by using the same value of 𝑥 to calculate the length of the line segment 𝐾𝐿 and then confirming that these values do indeed sum to 40. The expression for the length of 𝐾𝐿 is three 𝑥 minus two. So substituting 𝑥 equals three, we have three multiplied by three minus two. That’s nine minus two, which is equal to seven. Of course, 33 plus seven is equal to 40. So this confirms that our answer is correct. By forming and solving a quadratic equation then, we found that the length of the line segment 𝐽𝐾 is 33 units.
