Given that 𝑥 squared minus 10𝑥 is equal to 𝑥 plus 𝑝 all squared plus 𝑞, what are the values of 𝑝 and 𝑞?
To work out the values of 𝑝 and 𝑞, we will use the method of completing the square. This enables us to turn the general quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 into the equation 𝑥 plus 𝑝 all squared plus 𝑞, where the 𝑝 value can be calculated by dividing 𝑏 by two 𝑎 and 𝑞 is equal to 𝑐 minus 𝑏 squared divided by four 𝑎.
In our equation, 𝑥 squared minus 10𝑥, the value of 𝑎 is one as the coefficient of 𝑥 squared is one. In a similar way, the value of 𝑏 is negative 10 as the coefficient of 𝑥 is negative 10. The value of 𝑐 is zero as there is no free or loose term at the end of the equation.
Substituting these values into the equation 𝑝 equal 𝑏 divided by two 𝑎 gives us negative 10 divided by two multiplied by one. Two multiplied by one is two and negative 10 divided by two is negative five. Therefore, our value for 𝑝 is negative five. Substituting in the values of 𝑎, 𝑏, and 𝑐 into the equation 𝑞 equals 𝑐 minus 𝑏 squared divided by four 𝑎 gives us zero minus negative 10 squared divided by four multiplied by one.
Negative 10 squared is 100 and four multiplied by one is equal to four. And since 100 divided by four is 25, zero minus 25 is negative 25. Therefore, the equation 𝑥 squared minus 10𝑥 can be rewritten in the form 𝑥 minus 5 all squared minus 25. Well, our value for 𝑝 is negative five and our value for 𝑞 is negative 25.