Question 133011
 From the quadratic formula

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}


the discriminant consists of all of the terms in the square root. So the discriminant is


{{{D=b^2-4ac}}}


the discriminant tells us how many solutions (and what type of solutions) we can expect for any quadratic.



Now let's find the discriminant for {{{y=t^2+14t+49}}}:


{{{D=b^2-4ac}}} Start with the given equation


{{{D=(14)^2-4*1*49}}} Plug in a=1, b=14, c=49


{{{D=196-4*1*49}}} Square 14 to get 196


{{{D=196-196}}} Multiply -4*1*49 to get -196


{{{D=0}}} Combine 196 and -196 to get 0



Since the discriminant equals 0 , this means  there is one real solution. Remember if the discriminant is equal to zero, then the quadratic will have one real solution.




Notice if we graph {{{y=t^2+14t+49}}}, we can see that there is one real solution. So this verifies our answer.


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+14x+49) }}}