The solution to a quadratic equation is x is equals to 3 plus or minus square root of 4 minus 8p all over 4 where p is the element of Q.
Determine the value(s) of p that the roots of the equation are equal and non-real
x = ±√( (4-8p)/4)
square it
x^2 = (4-8p)/4
x^2 = 1 - 2p
x^2 + 2p = 1
x^2 + 2p + p^2 = 1+p^2 , I completed the square
(x+p)^2 = 1+p^2
x+p = ±√(1+p^2)
x = -p ± √(1+p^2)
since 1+p^2 is always positive , x will always be real
to have equal roots
-p + √(1+p^2) = -p - √(1+p^2)
2√(1+p^2) = 0
1+p^2 = 0
p^2 = -1
which is not possible, thus no equal roots
we could have seen that from x^2 + 2p - 1 = 0
for equal roots b^2 - 4ac = 0
4p^2 - 4(1)(-1) = 0
4p^2 = -4
p^2 = -1 , as above
To determine the values of p for which the roots of the quadratic equation are equal and non-real, we need to consider the discriminant of the quadratic equation.
The discriminant is calculated using the following formula: Δ = b^2 - 4ac, where Δ represents the discriminant, a, b, and c are coefficients of the quadratic equation.
In this case, we can see that the quadratic equation is in the form of ax^2 + bx + c = 0, with a = 1, b = 4 - 8p, and c = -4.
Substituting these values into the discriminant formula, we get:
Δ = (4 - 8p)^2 - 4(1)(-4)
= 16 - 64p + 64p^2 + 16
= 64p^2 - 64p + 32
For the roots to be equal and non-real, the discriminant Δ should be less than 0.
Therefore, we can solve the inequality 64p^2 - 64p + 32 < 0 to find the values of p.
Dividing every term by 32, we have:
2p^2 - 2p + 1 < 0
Next, we can solve this quadratic inequality by finding the values of p that satisfy the inequality.
One way to solve this is by factoring, but this inequality cannot be factored.
Another method is to use the quadratic formula:
The quadratic formula for an inequality is given by:
p = (-b ± √(b^2 - 4ac))/(2a)
Using the values from the inequality 2p^2 - 2p + 1 < 0:
a = 2, b = -2, and c = 1
p = (-(-2) ± √((-2)^2 - 4(2)(1)))/(2(2))
= (2 ± √(4 - 8))/(4)
= (2 ± √(-4))/(4)
= (2 ± 2i)/(4)
To find the values of p that satisfy the given inequality, we want the roots to be non-real. For a non-real root, we need to consider the ± 2i component.
Therefore, the values of p for which the roots of the quadratic equation are equal and non-real are:
p = (2 + 2i)/4 and p = (2 - 2i)/4.
In summary, the values of p for which the roots of the quadratic equation are equal and non-real are p = (2 + 2i)/4 and p = (2 - 2i)/4.