Find all positive integer values of $c$ such that the equation $x^2-7x+c=0$ only has roots that are real and rational. express them in decreasing order, separated by commas.

12, 10, 6 (descending order)

Step-by-step explanation:

has real and rational roots if is a nonnegative perfect square. is called the discriminant.

Let's calculate the discriminant of .

We want to be be positive or a zero perfect square.

So let's first solve the inequality:

Add on both sides:

Divide both sides be 4:

.

We know has to be a positive integer.

So we only need to evaluate for 12 numbers.

returns which is not a perfect square.

returns which is not a perfect square.

returns which is not a perfect square.

returns which is not a perfect square.

returns which is not a perfect square.

returns which is a perfect square.

returns which is not a perfect square.

returns which is not a perfect square.

returns which is not a perfect square.

returns which is a perfect square.

returns which is not a perfect square.

returns which is a perfect square.

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .