If the roots of the equation x2+a2=8x+6a are real, then a belongs to
I will assume you meant
x^2 + a^2 = 8x + 6a.
then
x^2 - 8x + a2 - 6a - 0
to have real roots, the discriminant ≥ 0
64 - 4(1)(a^2 - 6a) ≥ 0
64 - 4a^2 + 24a ≥ 0
a^2 - 6a - 16 < 0
(a-8)(a+2) ≤ 0
-2 ≤ a ≤ 8, a is a real number.
The given equation can be written as
x^2-8x+a-6a=0
Since the roots of the above equation are real
B^2-4ac>0
64-4(a^2-6a)>0
a^2-6a-16<0
(a+2)(a-8)<0
(-2,8)
To determine the range of values for 'a' for which the equation x^2 + a^2 = 8x + 6a has real roots, we can use the discriminant.
The discriminant (denoted by Δ) is a mathematical formula used to determine the nature of the roots of a quadratic equation. For the quadratic equation ax^2 + bx + c = 0, the discriminant is given by Δ = b^2 - 4ac.
In this case, the equation is x^2 + a^2 = 8x + 6a. By rearranging the terms, we can rewrite it as x^2 - 8x + a^2 - 6a = 0. Comparing this with the general quadratic equation form ax^2 + bx + c = 0, we have a = 1, b = -8, and c = a^2 - 6a.
Now, let's calculate the discriminant:
Δ = b^2 - 4ac
= (-8)^2 - 4(1)(a^2 - 6a)
= 64 - 4a^2 + 24a
= 4a^2 - 24a + 64
For the roots to be real, the discriminant Δ must be greater than or equal to zero (Δ ≥ 0).
Therefore, we solve the inequality:
4a^2 - 24a + 64 ≥ 0
To find the values of 'a' that satisfy this inequality, we can factor the quadratic expression or use the quadratic formula.
Using the quadratic formula, we have:
a = [ -(-24) ± √((-24)^2 - 4(4)(64)) ] / (2(4))
= [ 24 ± √(576 - 1024) ] / 8
= [ 24 ± √(-448) ] / 8
The discriminant (-448) is negative, which means the expression under the square root does not have real solutions. Therefore, there are no real values for 'a' that satisfy the inequality Δ ≥ 0.
In conclusion, there are no values of 'a' for which the equation x^2 + a^2 = 8x + 6a has real roots.