Quadratic equations, Vieta and Laws of Indices

A

1

Find the exact value of 1002² without using a calculator. 

1003998

2

Evaluate the following expression:

2² - 1² + 4² - 3² + ... 30² - 29²

Hint: Use difference of squares!

465

3

The symbolism ⌊x⌋ denotes the largest integer not exceeding x. For example, ⌊3⌋ = 3 and ⌊9/2⌋ = 4. Compute:

⌊sqrt(1)⌋ + ⌊sqrt(2)⌋ + ⌊sqrt(3)⌋ + ... ⌊sqrt(16)⌋ (AMC 10)

Hint: Try to evaluate this term by term. 

38

4

Find the value of x that satisfies the equation:

25^-2 = (5^48/x)/(5^26/x * 25^17/x) (AMC 10)

Hint: Rewrite the RHS as 5ⁿ, for some number n using laws of exponents

3

5

The parabolas y = ax² - 2 and y = 4 - bx² and intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area 12. What is a + b? (AMC 12)

Hint: Try to graph this situation. Find the y-intercepts and the roots in terms of a and b. 

1.5

B

1

If 2^7x * 2⁵ = 2²⁶, what is the value of x?

3

2

The vertex (the bottom/top point) of a parabola a(x - h)² + k is given by (h, k). Find the vertex of the parabola given by x² + 6x + 7.

Hint: Complete the Square!

(-3, -2)

3

There are two values of a for which the equation 4x² + ax + 8x + 9 = 0 has only one solution for x. What is the sum of those values of a? (AMC 10)

Hint: When does a quadratic have 1 solution?

-16

4

On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? (AMC 8)

Hint: Write an equation for this. 

28

5

We have a biased coin such that the probability of the coin landing on heads is p. Tim the timewaster wastes time to find out that the probability of flipping 1 head and 2 tails is 2 times the probability of flipping 2 consecutive heads. The value of p can be written as (a - sqrt(b))/c, where a, b, c are integers and the square root is fully simplified. Find a + b + c

Hint: Note that the probability of flipping a tails is 1-p

26

C

1

How many REAL solutions does the equation x² - 7x + 15 have?

0

2

Find the value of (2²⁰⁰¹ * 3²⁰⁰³)/6²⁰⁰² (AMC 10)

Hint: Can you rewrite the numerator to get a power of 6?

3/2

3

Come up in front of the class and sing the quadratic equation song. If you do this, you get 3 pieces of candy. 

Hopefully you sang it correctly. 

4

Find the sum of all solutions to the equation 3^{2x + 1} + 30(3^x) + 27 = 0. 

Hint: Does this look familiar? It kind of looks like a quadratic. How would we use quadratic techniques to solve?

2

5

Let P(x) be a quadratic polynomial with real coefficients satisfying x² - 2x + 2 <= P(x) <= 2x² - 4x + 3 for all real numbers x, and suppose P(11) = 181. Find P(16).

Hint: Complete the square of both the quadratics, and try to graph this. Where must P(x) be equal to the two quadratics?

406

D

1

Factor the expression a² - 10a + 21.

(a - 3)(a - 7)

2

The length and height of a square of side length x are increased by 4 and 5 respectively. As a result, the area of this new rectangle is 65 more than the area of the square. What is the area of the RECTANGLE?

Hint: Write an equation involving x. 

90

3

Let a and b be the two roots of x² - 9x + 7. Find (a - 1)(b - 1) WITHOUT finding the roots directly.

Hint: Expand (a - 1)(b - 1). Notice we have an ab and a + b term. How do we find this?

-1

4

Suppose that a and b are nonzero real numbers, and that the equation x² + ax + b = 0 has solutions a and b. Compute the pair (a, b). (AMC 10)

Hint: Use Vieta's formulas to write two equations for this situation. 

(1, -2)

5

Both roots of the quadratic equation x² − 63x + k = 0 are prime numbers. Compute the number of possible values of k. (AMC 10)

Hint: Write equations using Vieta's formula. Note that odd + odd = even and odd + even = odd, and that the roots are PRIME numbers.

1

E

1

Find the two solutions to the equation 2x² - 7x + 6 = 0.

x = 3/2, 2.

2

The expression (sqrt(sqrt(sqrt(x))) * x^-2)^-2/3 can be written as x^k. What is k?

Hint: Remember that sqrt(x) = x^1/2

5/4

3

How many different real numbers x satisfy the equation (x² - 5)² = 16? (AMC 8)

(Hint: Be careful with the square roots) 

4

4

Let x² + bx + c = 0 be a quadratic equation. Which of the following conditions will guarantee that this equation has a real solution:

(A) b is negative     (B) c is negative    (C) b is positive (D) c is positive      (E) None of these guarantee this

Hint: Think about the discriminant. 

B. c is negative

5

Let f(x) = sqrt(x + sqrt(x + sqrt(x ...))). How many values of k are there such that 0 <= k <= 100 and f(k) is an integer.

Hint: Let n = sqrt(x + sqrt(x + sqrt(x ...))). Can we write a quadratic equation for n? 

10