If alpha and beta are zeroes of 4x-5x-1. Find the value of alpha square multiplied by beta
To find the value of α² * β, we first need to find the values of α and β.
Given that α and β are the zeroes of 4x^2 - 5x - 1, we can use the quadratic formula to find their values.
The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the solutions are:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4, b = -5, and c = -1.
First, let's calculate the discriminant, which is inside the square root:
Δ = b^2 - 4ac
Δ = (-5)^2 - 4(4)(-1)
Δ = 25 + 16
Δ = 41
Since the discriminant is positive, we have two distinct real roots.
The solutions for α and β can be found by substituting the values into the quadratic formula:
α = (-(-5) + √41) / (2(4))
α = (5 + √41) / 8
β = (-(-5) - √41) / (2(4))
β = (5 - √41) / 8
Now, let's calculate α² * β:
α² * β = [(5 + √41) / 8]² * [(5 - √41) / 8]
= [(5 + √41) * (5 + √41)] / [8 * 8] * [(5 - √41) / 8]
= [(25 + 2 * 5 * √41 + 41) / 64] * [(5 - √41) / 8]
= [(66 + 10 * √41) / 64] * [(5 - √41) / 8]
= (330 - 64 * √41 + 50 * √41 - 410) / (64 * 8)
= (-80 - 14 * √41) / 512
Therefore, the value of α² * β is (-80 - 14 * √41) / 512.
To find the value of α² * β, we need to first find the values of α and β.
Given that α and β are the zeroes of the quadratic equation 4x² - 5x - 1 = 0, we can use the quadratic formula to find their values.
The quadratic formula states that for a quadratic equation of the form ax² + bx + c = 0, the solutions (or zeroes) can be found using the formula:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 4, b = -5, and c = -1.
Using the quadratic formula, we can find the values of α and β:
For α:
x = (-(-5) ± √((-5)² - 4 * 4 * (-1))) / (2 * 4)
= (5 ± √(25 + 16)) / 8
= (5 ± √41) / 8
So, α = (5 + √41) / 8 or α = (5 - √41) / 8.
For β:
x = (-(-5) ± √((-5)² - 4 * 4 * (-1))) / (2 * 4)
= (5 ± √(25 + 16)) / 8
= (5 ± √41) / 8
So, β = (5 + √41) / 8 or β = (5 - √41) / 8.
Now that we have the values of α and β, we can find the value of α² * β:
α² * β = ((5 + √41) / 8)² * ((5 - √41) / 8)
= (25 + 2 * 5 * √41 + 41 / 8) * ((5 - √41) / 8)
= (25 + 10√41 + 41 / 8) * ((5 - √41) / 8)
= (25/64) + (5/4) * √41 + (41/64) * (5 - √41)
= (25/64) + (5/4) * √41 + 205/64 - (41/64) * √41
= (25/64) + 205/64 + (5/4 - 41/64) * √41
= (25/64) + 205/64 + (320/64 - 41/64) * √41
= (25 + 205 + 279) / 64
= 509 / 64
Therefore, α² * β = 509 / 64.
Given equation is 4x2-5x-1,now a+b=-(-5)/4=a=5-4a/4 again a.b=-1/4 put the value of a we get a=1,and b=1/4 so a2b=1/4.
Assuming you mean:
4x^2 - 5x - 1 = 0
x = (5 ± √41)/8
so alpha = (5+√41)/8 and beta = (5-√41)/8
a^2b = (5+√41)/8 * (5+√41)/8 * (5-√41)/8
= (5+√41)/8 * (25 - 41)/64
= (5+√41)/8 * (-16/64)
= (-5-√41)/32
GAURAV concluded that a = 1, b = 1/4
but those are not the zeros of the given equation.
my answer can be verified with a calculator.