if g(x)=kx^2...contniued in details?

Question

if g(x)=kx'^2-5x+25 has only one distinct root, what is the value of k?
a. 1/25
b. 1/16
c. 1/10
d. 1/5
e. 1/4

oh and can anyone explain to me what the "one distinct root" is/means?

Answer 1

This means the discriminant equals zero or:
sqrt(b^2 - 4ac) = 0
b^2 -4ac = 0
b^2 = 4ac
a = k = b^2 / 4c = 25 / (4*25) = 1/4

Answer 2

A quadratic equation has the canonical form

ax^2 + bx + c = 0

It has roots (given by the quadratic formula) of

x = (-b + sqrt(b^2 - 4ac))/2a

and

x = (-b - sqrt(b^2 - 4ac))/2a

The term (b^2 - 4ac) is called the 'discriminent' and the solution set for the quadratic can be one of three things depending on its value

If (b^2 - 4ac) is positive, there will be 2 distinct roots
If (b^2 - 4ac) is zero, there will be one distinct root.
If (b^2 - 4ac) is negative, there will be 2 complex roots

In your problem, the ask for the case where the discriminent is zero, so write

0 = (-5)^2 - 4*k*25

and solve for k (which must be 1/4)

The term 'distinct root' is used to differentiate from a 'repeated' root. For example (x-1)^n has n roots which are all at 1 (this is a classic 'repeated root'). Another example might be the fraction

(x^2 - 4)/(x^2 - 5x + 6)

If you are looking for roots (where the function goes to 0) it's pretty tempting to just say that they exist where the numerator goes to zero. That is at x = 2 and x = -2. But you have to be careful because the *denominator* also goes to zero at x = 2 (and at x = 3) and so the function fails to exist at x = 2 (it would be 0/0 which is a big time no no )

In this case the 'distinct root' is at x = -2.

Hope that helps.


Doug

Answer 3

One distinct root means that the two roots of the equation are identical. Like, (nx-5)*(nx-5)=(nx)^2-10x+25 has one distinct root. Since you know that the equation must be in this form (since "25" is the last term), and both terms are identical, you just have to choose a coefficient of x so that it changes the -10x to -5x in the middle of the equation. That is, a factor of 1/2. And once you square it, the answer becomes obvious.

Answer 4

The function g(x) has two roots, since it is a quadratic. But, since it has "one distinct root", the two roots are the same number, also called having "multiplicity two" For that situation to occur, the quadratic has to a be a perfect square:

g(x)= kx^2 - 5x + 25
= (sqrt(k)*x - sqrt(25) )^2

But, 2 * -sqrt(25) * sqrt(k) = -5, for the quadratic to be a perfect square:
2 * -5 * sqrt(k) = -5
= -10*sqrt(k) = -5
sqrt(k) = 1/2
k = 1/4

or "E"

Answer 5

Answer would be "e"

One distinct root means that there is only one point at which that function equals zero.
If you are familiar with the general formula, for having only one distinct root, you need b^2-4ac=0

Answer 6

g(x) = kx^2 - 5x + 25

b^2 - 4ac = 0

(-5)^2 - 4(k)(25)
25 - 100k = 0
-100k = -25
100k = 25
k = 25/100
k = 1/4

ANS : e.) 1/4

b^2 - 4ac = 0 gives you one distinct root
b^2 - 4ac = +n gives you 2 real roots
b^2 - 4ac = -n gives you 2 complex(imaginary) roots

Answer 7

one distinct root means one and only one solution/value of x. That is, that the graph cuts the x axis once.
and its not a repeated root, which means that the interception is not a turning point

Answer 8

10 particular solutions because 100 short solutions may be about 100 distinct issues and also you do not knw the position the fellow is comming from. I continuously provide causes in the back of my solutions because I merely ought to describe myself