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Find the range of values of k which the equation 3x2-3kx plus k2 is always positive?
3x2 - 3kx + k2 > 0, k = ? (a = 3, b = -3k, c = k2)
The parabola opens upward (a > 0), so we have a minimum point at the
vertex = (- b/2a, c - b2/4a) = (- -3k/6, k2 - (-3k)2/12) = (k/2, k2/4).
Since the y-coordinate of the vertex is always a positive value, except when k = 0, then the x-coordinate could have a positive or negative value.
So that, the parabola would lie above the x-axis for all values of k, except when k is zero.
Thus, the equation 3x2 - 3kx + k2 > 0, for k ≠ 0.
Or
Complete the square:
3x2 - 3kx + k2 (divide by 3 all the terms)
= x2 - kx + k2/3
= [x2 - kx + (k/2)2]+ k2/3 - (k/2)2
= (x - k/2)2 + k2/3 - k2/4
= (x - k/2)2 + k2/12
So the equation (x - k/2)2 + k2/12 represents the translation of x2, k/2 units right when k > 0, or k/2 units left when k < 0, and k2/12 units up.
Thus, for k ≠ 0 the given equation is always positive.
What else can I help you with?
Can probability values be negative?
No. Probability values always have to be positive.
Can you solve this polynomial equation by factorization and examine the curvature of the function the equation function is X squared plus 4x minus 5 equals 0?
For the equation: x2+4x-5 = 0 Factoring yields: (x+5)(x-1) = 0 The zero identity states that setting these factors individually equal to zero will yield valid solutions to this equation. x+5 = 0, x = -5 x-1 = 0, x = 1 So the values x=-5 and x=1 solve this equation. For the mathematical function: y = x2+4x-5 The derivatives for this function are as such: y' = 2x+4 y'' = 2 The second derivative is a constant positive number (2), so it is always positive. Since the second derivative is always positive, its curvature is always concave upward.
What is an equation that's true for all values?
an equation that's true for all values is an identity.
Do absolute values always have two answers?
An absolute value doesn't have 2 answers. It isn't even a question. It's always positive.
Is it possible for an equation to have more than one correct solution?
Definitely.The equation [ x^2 = 4 ] has two solutions.x = +2x = -2The square root of any number can be a positive number or its negative. The solution for a quadratic equation often has two different values. However having two different values is still a single solution.
Can probability values be negative?
No. Probability values always have to be positive.
What is an equation that is true regardless of the values of the variables called?
An equation that is always true is an identity.
What are the possible values of the equation a is a multiple of 2 less than 10?
For positive numbers 8,6,4,2
Can absolute value be negative?
No, absolute values are always positive.
Are all probability values always positive numbers?
yes.
If an equation is an identity what is true about the solution?
It is true for all permissible values of any variables in the equation. More simply put, it is always true.
When you calculate a mean the sign of the quotient always depends on the sign of the dividend why?
The mean is calculated by dividing the sum of a set of values (the dividend) by the number of values in that set (the divisor). The sign of the quotient, or the mean, will depend on the sign of the dividend because the divisor (the count of values) is always positive. If the sum of the values is positive, the mean will also be positive; if the sum is negative, the mean will be negative. Thus, the sign of the mean directly reflects the sign of the sum of the values.
Will an absolute value be graphed on the positive or negative side of the number line?
Absolute values are always positive; so graph it on the positive side of the number line.
Can you solve this polynomial equation by factorization and examine the curvature of the function the equation function is X squared plus 4x minus 5 equals 0?
For the equation: x2+4x-5 = 0 Factoring yields: (x+5)(x-1) = 0 The zero identity states that setting these factors individually equal to zero will yield valid solutions to this equation. x+5 = 0, x = -5 x-1 = 0, x = 1 So the values x=-5 and x=1 solve this equation. For the mathematical function: y = x2+4x-5 The derivatives for this function are as such: y' = 2x+4 y'' = 2 The second derivative is a constant positive number (2), so it is always positive. Since the second derivative is always positive, its curvature is always concave upward.
Why is the domain of a parabola all real numbers?
The domain of a parabola is always all real numbers because the domain represents the possible x values. The x values are shown on the horizontal axis or x axis. Because, in a parabola, the 2 sides of the parabola go infinitely in a positive or negative direction, there is always a y value for any x value that u plug in to the equation.
What does solve the equation mean?
Find values for the variable that satisfy the equation, that is if you replace those values for the variable into the original equation, the equation becomes a true statement.
What is an equation that's true for all values?
an equation that's true for all values is an identity.
