Video Transcript
Given that the equation 𝑥 squared minus negative two 𝑚 plus 28𝑥 plus 𝑚 squared equals zero has no real roots, find the interval that contains 𝑚.
We recall that any quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero that has no real roots will have a discriminant 𝑏 squared minus four 𝑎𝑐 less than zero. In this question, we begin by rewriting our equation by distributing the parentheses so that 𝑥 squared plus two 𝑚 minus 28𝑥 plus 𝑚 squared is equal to zero. The value of 𝑎, the coefficient of 𝑥 squared, is equal to one. The coefficient of 𝑥, 𝑏, is equal to two 𝑚 minus 28. The constant term 𝑐 is equal to 𝑚 squared.
Substituting these values, we have two 𝑚 minus 28 squared minus four multiplied by one multiplied by 𝑚 squared is less than zero. To square two 𝑚 minus 28, we multiply two 𝑚 minus 28 by two 𝑚 minus 28. We can then distribute the parentheses or expand the brackets using the FOIL method. Multiplying the first terms gives us four 𝑚 squared. Both the outer and inner terms have a product of negative 56𝑚. The last terms, negative 28 and negative 28, have a product of 784.
Two 𝑚 minus 28 squared is therefore equal to four 𝑚 squared minus 112𝑚 plus 784. This means that our inequality becomes four 𝑚 squared minus 112𝑚 plus 784 minus four 𝑚 squared is less than zero. The four 𝑚 squareds cancel. We can then add 112𝑚 to both sides so that 784 is less than 112𝑚. Dividing through by 112, we get seven is less than 𝑚 or 𝑚 is greater than seven.
We are asked to write this using interval notation. 𝑚 therefore belongs to the set of values greater than seven and less than ∞. We use the curved brackets or parentheses and not square ones as we cannot include the values seven or ∞. The equation 𝑥 squared minus negative two 𝑚 plus 28𝑥 plus 𝑚 squared is equal to zero has no real roots when 𝑚 lies between seven and ∞.
