Answers to

Math Challenge Problems

Challenge Question 1The "UPCAT-level" problem that can be solved using

the discriminant:

The equation ax2 + 2x + c = 0 has two distinct

RATIONAL roots, where a and c are both rational.

Which of the following is a possible value for ac?

The condition “where a and c are both rational” is added to avoid ambiguity.

CQ1 Solution

Compute the discriminant. In this case, b = 2:

2 24 2 4

4 4

b ac ac

ac

Note that

The equation will have distinct RATIONAL roots if

the discriminant D = b2 4ac is a PERFECT

SQUARE.

4 4 4 1ac ac

CQ1 Solution

Since 4 is a perfect square, 1 – ac MUST also be a

perfect square so that is a perfect

square.

Among the choices, ac = 3 is a possible value

since 1 – (– 3) = 4 is a perfect square.

Therefore, the answer is (b).

4 1 ac

CQ1 Solution

QUESTION: If ac = 1, 1 – 1 = 0 is also a perfect

square. Why is this NOT allowed to be a value of

ac?

CLUE: Read the question again!

CQ1 Solution

If ac = 1, 1 – 1 = 0 is also a perfect square. BUT, it

will make D = 0, which means that the equation

will have EQUAL ROOTS.

Now, read the question again. The given equation

must have DISTINCT rational roots, which means

D > 0. Kaya dapat, ang conditions ay:

1. D is a perfect square

2. D > 0.

CHALLENGE: What is the value of ?

Challenge Question 2

5 4 2a b c d 2

3 3 3

HINT: What must be the power of 8 to have 16?

Using the definition of the logarithm (see the

solution to Q11 in the CEER presentation):

CQ 2 Solution

8log 16 8 16yy

At this point, you can SUBSTITUTE the choices to

the exponent y. The answer is 4/3 since 4

34/ 3 48 8 2 16

Here’s the formal

solution. The idea is

to make each side

have the same

smallest possible

base, and equate

the exponents to

solve for y.

CQ 2 Solution

3 4

3 4

8 16

2 2

2 2

3 4

4

3

y

y

y

y

y

Dakal a Salamat!