Determine for what value(s) of d the quadratic equation 5x^2-10x+d = 0 has
i) real and distinct roots
ii)real and equal roots
iii)non-real roots
This is what I did to solve for A.
5x^2-10x+d = 0
b^2-4ac = 0
(-10)^2-4(5)(d) = 0
100-20d = 0
-20d = -100
d = 5
Need help with others.
Close. Actually, you solved for (ii).
i) D > 0
100 - 20d > 0
d < 5
ii) D=0
100-20d = 0
d = 5
iii) D < 0
100-20d < 0
d > 5
To determine the values of "d" for which the quadratic equation 5x^2-10x+d = 0 has (i) real and distinct roots, (ii) real and equal roots, or (iii) non-real roots, we need to analyze the discriminant (b^2-4ac) of the equation.
The quadratic equation can be written as ax^2 + bx + c = 0, where a = 5, b = -10, and c = d.
(i) Real and Distinct Roots:
For a quadratic equation to have real and distinct roots, the discriminant (b^2-4ac) must be greater than zero.
In this case,
Discriminant = (-10)^2 - 4(5)(d)
= 100 - 20d
To find the values of "d" that satisfy this condition, we set the discriminant greater than zero and solve:
100 - 20d > 0
Solving the inequality:
100 > 20d,
5 > d
So, for "d" values less than 5, the quadratic equation has real and distinct roots.
(ii) Real and Equal Roots:
For a quadratic equation to have real and equal roots, the discriminant (b^2-4ac) must be equal to zero.
In this case,
Discriminant = (-10)^2 - 4(5)(d)
= 100 - 20d
To find the values of "d" that satisfy this condition, we set the discriminant equal to zero and solve:
100 - 20d = 0
Solving the equation:
100 = 20d,
5 = d
So, for "d" equals 5, the quadratic equation has real and equal roots.
(iii) Non-real Roots:
For a quadratic equation to have non-real (complex) roots, the discriminant (b^2-4ac) must be less than zero.
In this case,
Discriminant = (-10)^2 - 4(5)(d)
= 100 - 20d
To find the values of "d" that satisfy this condition, we set the discriminant less than zero and solve:
100 - 20d < 0
Solving the inequality:
100 < 20d,
5 < d
So, for "d" values greater than 5, the quadratic equation has non-real roots.
In summary:
- For "d" values less than 5, the quadratic equation has real and distinct roots.
- For "d" equal to 5, the quadratic equation has real and equal roots.
- For "d" values greater than 5, the quadratic equation has non-real (complex) roots.