Mathematics, 01.08.2019 10:30 abadir2008

Show that the equation 2x^2+5hx=3k has real and distinct roots for all positive values of k.

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The given equation is quadratic and nature of roots of quadratic equation are dependant on its discriminant. Writing the equation in standard form:

$2 x^{2} +5hx-3k = 0$

Finding the discriminant:

$Discriminant = (5h)^{2} - 4(2)(-3k) = (5h)^{2}+24k$

The roots of a quadratic equation are real and distinct if the value of its discriminant is greater than 0.

If we observe the value of discriminant, it is a sum of two terms in this case. The first term is square of (5h), which will always be a positive term or zero if h is zero. The second term is 24k. The value of 24k will always be positive for positive values of k.

Thus for positive values of k, the discriminant of given equation is always positive and thus we can say the root of given equation will always be real and distinct for positive values of k.

Answer from: Quest

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1

Answer from: Quest

it is not that hard all u have to do is plot the graph

step-by-step explanation:

Need with this if u can answer the question comment it

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Show that the equation 2x^2+5hx=3k has real and distinct roots for all positive values of k....