One of the roots of the equation (x)x + kx - 6 = 0 is 3, and k is a constant.
Quantity A : The value of k
Quantity B : -1Quantity A is greater., Quantity B is greater., The two quantities are equal., The relationship cannot be determined from the information given.
5 Explanations
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Elizabeth Mendes
I understand how you got each of the factors, but do you know that K is -1 because the terms in the factor added together = -1?
And plugging in 3 directly to the equation is an alternate way to solve?
TO address your first question, we get -1 for K by applying the FOIL method to (x-3)*(x+2). This gets us x^2 + 2x - 3x - 6. 2x - 3x = -1x, so yes, the x terms in the factor combine to make -1.
You're also correct that plugging 3 in for x and solving for k will also get you -1:
My answer was that the relationship can not be determined. I plugged in 3 for x to get - 1 for K, but then I reasoned that the prompt says that ONE of the roots of the equation is 3, but couldn't the root be -3 , -2 , or +2 ? Wouldn't that change the value of K when you plug those in?
A single equation can have more than one root - but that doesn't change the equation itself. You can think of the roots of an equation as the values of x that will make the equation equal 0.
Here we know that k is -1, so the equation is y=x^2-x-6. We know that one of the roots of the equation is 3, and if you plug (x=3) into the equation, then y=0. The other root is (-2), because when (x=-2), y=0. So even though this equation has two different roots, the value of k never changes.
In these equations, do you just assume that +kx is just the form of the generalized way of writing the equation, so the plus in front of k is not indicative of the actual sign of what k is equal to? ie x^2 - x - 6.
Yes, that's right. The operation (addition, subtraction, multiplication, etc.) does not signify whether the value of a variable is positive or negative. So the equation says " + kx", but if either k or x is a negative number, the over value is DECREASED by that term, because we are then adding a negative number, which is the same as subtracting. Does that make sense?
Hi,
No -- (x - 3) and (x + 2) are the factors of the polynomial.
The ROOTS of the polynomial are the values of the variable that make the polynomial equal to zero.
x = 3 is a root because is makes the polynomial zero.
5 Explanations