The lengths of the sides of quadrilateral PQRS are PQ=x^2, QR=20-x, RS=x^2-4, PS=1-4x. Find a value of x that will make PQRS a kite (two pairs of congruent consecutive sides).

which two points are the endpoints of the short arm, and which two the endpoints of the long arm?

set short/long sides equal and solve for x. What do you get?

The points of PQRS go around clockwise. So x=4?

that doesn't answer my question. Where is P?

To find the value of x that will make quadrilateral PQRS a kite, we need to find two pairs of congruent consecutive sides.

A kite has two pairs of congruent consecutive sides, meaning that PQ = RS and QR = PS.

Let's set up the equations:

1) PQ = RS

If we substitute the expressions for PQ and RS, we get:
x^2 = x^2 - 4

Simplifying this equation, we find:
0 = -4

However, this equation has no solution. Therefore, PQ is not equal to RS and cannot form a pair of congruent consecutive sides for the kite.

2) QR = PS

Substituting the expressions for QR and PS, we get:
20 - x = 1 - 4x

Simplifying this equation, we find:
3x = 19

Dividing both sides by 3, we get:
x = 6.33 (rounded to two decimal places)

Therefore, the value of x that will make PQRS a kite is approximately 6.33.