The quadratic formula, has a part we call the "discriminant" defined by the variables that are inside the square root, and is denotated by "delta":
Δ=b2−4acWhenever we solve a quadratic equation that is complete and we analyze the discriminant, we can get 3 scenarios:
if→Δ>0=>∃x1,x2/ax2+bx+c=0This just means: "if the discriminant is greater than zero, there will exist two x-intercepts"
And for the second scenario:
if→Δ=0→∃xo/ax2+bx+c=0This means: "if the discriminant is equal to zero, there will be one and only one x-intercept"
And for the last scenario:
if→Δ<0→∃x∉R/ax2+bx+c=0This means that :"if the discriminant is less than zero, there will be no x-intercepts"
So, if we take your excercise and analyze the the discriminant:
3x2+7x+m=ywe will find the values that satisfy y=0 :
3x2+7x+m=0And we'll analyze the discriminant:
Δ=72−4(3)(m)And we are only interested in the values that make the discriminant equal zero:
72−4(3)(m)=0All you have to do is solve for "m".
