Given that 𝑥 squared plus two 𝑥 plus five is equal to 𝑥 plus 𝑝 all squared plus 𝑞, what are the values of 𝑝 and 𝑞?
Well, our method of completing the square allows us to write any quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equal to 𝑎 multiplied by 𝑥 plus 𝑝 all squared plus 𝑞, where 𝑝 is equal to 𝑏 divided by two 𝑎 and 𝑞 is equal to 𝑐 minus 𝑎𝑝 squared.
For the example 𝑥 squared plus two 𝑥 plus five, our values of 𝑎, 𝑏, and 𝑐 are positive one, positive two, and positive five, respectively. Substituting the values of 𝑏 and 𝑎 into the equation 𝑝 equals 𝑏 divided by two 𝑎 gives us 𝑝 is equal to two divided by two multiplied by one. This is equal to one as two multiplied by one is two and two divided by two equals one.
We can use the equation 𝑞 equals 𝑐 minus 𝑎𝑝 squared to work out the value of 𝑞 in this case. Five minus one multiplied by one squared is equal to four. Therefore, 𝑝 is equal to one and 𝑞 is equal to four. This enables us to rewrite the quadratic equation 𝑥 squared plus two 𝑥 plus five as 𝑥 plus one all squared plus four.