Mastering Algebra for AMC 10/12

Set up a system of equations with speeds x, x+5, x+10 and hours h, h+1, h+2 to solve 2008 AMC 10A problem 15 and show Jan drove 150 more miles.

Apply a system of equations to solve the 2010 AMC 12B problem, modeling rain and sun speeds (30 mph and 20 mph) to determine 24 minutes of rain.

Solve the 2011 AMC 10B problem by squaring, using a = |x| to get a quadratic, factor to |x|=8, and deduce x=±8, yielding 64.

Explore inequalities as ranges defined by greater than, less than, and greater or equal to. Learn solving tricks, including sign flips when dividing by negatives and using a number line.

Solve a 2014 AMC 10A inequality problem, determine which of four statements must be true, and show that only the first inequality x + y < a + b holds.

Solve the 2010 AMC 10A problem 11 by converting the inequality a ≤ 2x+3 ≤ b with length 10 into an interval, then deduce B minus A equals 20.

Count the number of ordered pairs (x, y) of positive integers with x ≤ y ≤ 60 and y ≤ 2x, using symmetry and case analysis, yielding 480.

Explore how percent values are decimals or fractions over one hundred, including zero point one percent equals one over thousand, and compute percent change and 60 percent of 300.

Compute year-to-year percentage increases for the 2007 AMC 10 problem 6, using (new minus old) over old, and identify the largest rise, which occurs from 2002 to 2003.

Master exponents and radicals by applying core rules: product, quotient, power, power of a power, negative exponents with reciprocals, zero exponents, and nth roots.

Rewrite bases as powers of 3 and 5 for the 2007 AMC 10 problem 9, equate exponents to find a = -12 and b = -5, then ab = 60.

The lecture walks through using exponent rules to simplify powers of five, convert twenty-five to five squared, and solve for x, yielding x = 3.

Let a = sqrt(49 − x^2) and b = sqrt(25 − x^2). With a − b = 3, apply the difference of squares to find a + b = 8.

Explore special factorizations used on the AMC, including completing the square and Simon's favorite factoring trick, and solve quadratics and polynomials with explicit examples.

Use special factorization tricks to find the two a values for which the equation x^2+9x+9=0 has a single solution, and compute their sum as -16.

Solve an AMC problem on two primes using special factorization, rewriting pq minus p minus q as (p minus 1)(q minus 1) to find the pair yielding 119.

Using a special factorization, the lecture rewrites the problem as (3x^2+1)(y^2-10). It then finds x^2=4 and y^2=49, yielding 588.

Examine arithmetic and geometric sequences, focusing on constant difference and ratio, finite and infinite sums, and nth-term formulas with examples like 1,3,5,7 and 1,2,4,8.

Describe five consecutive terms of an arithmetic sequence whose sum is 30, and use central-term symmetry to deduce C = 6.

Identify how many values of K produce 2005 in the sequence by requiring divisibility of 2004, then factor 2004 to obtain twelve as the divisor count.

Solve the 2016 AMC 10B problem 16 by using an infinite geometric series with second term one. Maximize r(1 - r) to minimize S, yielding S = 4.

Learn Vieta's formulas for polynomials, using coefficients to find the sum and product of roots, and apply quadratic and cubic examples to practice these relationships.

Solve 2006 AMC 10B problem 14 using ETA formulas and root relations A and B, with A+1/B and B+1/A as roots, to find q = 9/2.

Solve the 2003 AMC 10A problem 18 by forming a quadratic and applying Viète’s formulas to find the sum of the reciprocals of the roots, yielding -1.