if 2x^2-(b-4)x-4(b+2)=0
has equal root then.the value of b?
if two roots are identical
then
(b-4)^2 - 4(2)(-4)(b+2) = 0
b^2 - 8 b + 16 = -32b - 64
b^2 + 24 b + 80 = 0
(b+4)(b+20) = 0
Well, if the equation has equal roots, it means that the discriminant is zero. So let's calculate it:
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
In this case, the equation is 2x^2 - (b-4)x - 4(b+2) = 0.
So, b^2 - 4(2)(-4)(b+2) = 0.
Simplifying the expression, we have b^2 + 32(b+2) = 0.
Expanding further, we get b^2 + 32b + 64 = 0.
Now, to find the value of b, we can factorize the equation: (b+16)(b+4) = 0.
This means that either (b+16) = 0 or (b+4) = 0.
So, the possible values of b are -16 or -4.
But hey, I'm a Clown Bot, and this math stuff really makes my circuits tingle! You might want to double-check those calculations just to be sure!
To find the value of b when the equation 2x^2 - (b-4)x - 4(b+2) = 0 has equal roots, we can use the discriminant. The discriminant is given by the formula: b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In this case, a = 2, b = -(b-4), and c = -4(b+2). Let's substitute these values into the discriminant formula:
Discriminant = (-b+4)^2 - 4(2)(-4b-8)
Simplifying further:
Discriminant = (b^2 - 8b + 16) - (-32b - 64)
= b^2 - 8b + 16 + 32b + 64
= b^2 + 24b + 80
For the equation to have equal roots, the discriminant must equal zero. Therefore, we can set the discriminant equal to zero and solve for b:
b^2 + 24b + 80 = 0
Unfortunately, this equation cannot be factored easily, so we can use the quadratic formula:
b = (-24 ± √(24^2 - 4(1)(80))) / (2)
Simplifying further:
b = (-24 ± √(576 - 320)) / (2)
b = (-24 ± √256) / 2
b = (-24 ± 16) / 2
We have two possible values for b:
1. b = (-24 + 16) / 2 = -8
2. b = (-24 - 16) / 2 = -20
So, the two possible values for b when the equation has equal roots are -8 and -20.
To find the value of b for which the given quadratic equation has equal roots, we can use the discriminant. The discriminant (represented by the symbol Δ) is a mathematical expression that helps determine the nature of the roots.
For a quadratic equation of the form ax^2 + bx + c = 0, the discriminant (Δ) is given by the formula:
Δ = b^2 - 4ac
If the discriminant is equal to zero (Δ = 0), then the quadratic equation has equal roots.
In the given equation, 2x^2 - (b - 4)x - 4(b + 2) = 0, the coefficients are:
a = 2
b = -(b - 4) = -b + 4
c = -4(b + 2) = -4b - 8
Using the coefficients, we can find the discriminant as follows:
Δ = (-b + 4)^2 - 4(2)(-4b - 8)
Expanding the terms:
Δ = b^2 - 8b + 16 - 32(-b -8)
Δ = b^2 - 8b + 16 + 32b + 256
Δ = b^2 + 24b + 272
To find the value of b that gives equal roots, we equate the discriminant to zero:
Δ = 0
b^2 + 24b + 272 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
By factoring, we need to find two numbers that multiply to 272 and add up to 24. The numbers are 16 and 17:
(b + 16)(b + 17) = 0
Setting each factor equal to zero gives us:
b + 16 = 0 OR b + 17 = 0
Solving each equation, we get:
b = -16 OR b = -17
Therefore, the equation 2x^2 - (b - 4)x - 4(b + 2) = 0 will have equal roots when b is either -16 or -17.