Video Transcript
A particle started moving along a straight line. At time 𝑡, where 𝑡 is greater than or equal to zero, its position relative to a fixed point is given by 𝑟 is equal to two 𝑡 squared plus four 𝑡 minus two. Determine all the possible values of 𝑡 at which 𝑟 equals four.
In this question, we are given an equation for the position 𝑟 of a particle in terms of time 𝑡. We’re asked to calculate the values of 𝑡 at which 𝑟 equals four. Replacing 𝑟 with four, we have four is equal to two 𝑡 squared plus four 𝑡 minus two. We can subtract four from both sides of this equation so that we have a quadratic that is equal to zero. Two 𝑡 squared plus four 𝑡 minus six is equal to zero.
Dividing both sides of this equation by two, we have 𝑡 squared plus two 𝑡 minus three equals zero. Our next step is to factor the right-hand side of our equation. The first term in each set of parentheses is 𝑡 as 𝑡 multiplied by 𝑡 is 𝑡 squared. Our next step is to find a pair of numbers that have a sum of positive two and a product of negative three. These are negative one and three as negative one plus three is equal to positive two and negative one multiplied by three is negative three. Our two sets of parentheses are 𝑡 minus one and 𝑡 plus three.
As the product of these equals zero, either 𝑡 minus one equals zero or 𝑡 plus three equals zero. These equations give us values of 𝑡 equal to one and negative three. As we are dealing with time and 𝑡 must be greater than or equal to zero, we can eliminate 𝑡 equals negative three. The only value of 𝑡 at which 𝑟 equals four is one.
