Answer:
Step-by-step explanation:
Our equations are
[tex]y = -3x^2 + x + 12\\y = 2x^2 - 6x + 5\\y = x^2 + 7x - 11\\y = -x^2 - 8x - 16\\[/tex]
Let us understand the term Discriminant of a quadratic equation and its properties
Discriminant is denoted by D and its formula is
[tex]D=b^2-4ac\\[/tex]
Where
a= the coefficient of the [tex]x^{2}[/tex]
b= the coefficient of [tex]x[/tex]
c = constant term
Properties of D: If D
i) D=0 , One real root
ii) D>0 , Two real roots
iii) D<0 , no real root
Hence in the given quadratic equations , we will find the values of D Discriminant and evaluate our answer accordingly .
Let us start with
[tex]y = -3x^2 + x + 12\\a=-3 , b =1 , c =12\\D=1^2-4*(-3)*(12)\\D=1+144\\D=145\\D>0 \\[/tex]
Hence we have two real roots for this equation.
[tex]y = 2x^2 - 6x + 5\\[/tex]
[tex]y = 2x^2 - 6x + 5\\a=2,b=-6,c=5\\D=(-6)^2-4*2*5\\D=36-40\\D=-4\\D<0\\[/tex]
Hence we do not have any real root for this quadratic
[tex]y = x^2 + 7x - 11\\a=1,b=7,-11\\D=7^2-4*1*(-11)\\D=49+44\\D=93\\[/tex]
Hence D>0 and thus we have two real roots for this equation.
[tex]y = -x^2 - 8x - 16\\a=-1,b=-8,c=-16\\D=(-8)^2-4*(-1)*(-16)\\D=64-64\\D=0\\[/tex]
Hence we have one real root to this quadratic equation.
