Given a quadratic equation of the form:
[tex]ax^2+bx+c=0[/tex]
The discriminant is:
[tex]D=b^2-4ac[/tex]
And we can know the number of solutions with the value of the discriminant:
• If D < 0, the equation has 2 imaginary solutions.
,
• If D = 0, the equation has 1 real solution
,
• If D > 0, the equation has 2 real solutions.
Equation One:
[tex]x^2-4x+4=0[/tex]
Then, we calculate the discriminant:
[tex]D=(-4)^2^-4\cdot1\cdot4=16-16=0[/tex]
D = 0
There are 1 real solution.
Equation Two:
[tex]-5x^2+8x-9=0[/tex]
Calculate the discriminant:
[tex]D=8^2-4\cdot(-5)\cdot(-9)=64-20\cdot9=64-180=-116[/tex]
D = -116
There are 2 imaginary solutions.
Equation Three:
[tex]7x^2+4x-3=0[/tex]
Calculate the discriminant:
[tex]D=4^2-4\cdot7\cdot(-3)=16+28\cdot3=16+84=100[/tex]
D = 100
There are 2 real solutions.
Answers:
Equation 1: D = 0, One real solution.
Equation 2: D = -116, Two imaginary solutions.
Equation 3: D = 100, Two real solutions.
