Math · Advanced Math foundations
Advanced Math drill
22 quick questions across 8 foundational sub-topics, starting from the basics. Read the mini-lesson and worked example first, then try the practice questions. Every answer checks instantly.
Mini-lesson
Factoring is undoing multiplication. For x² + bx + c, find two numbers that multiply to c and add to b, then write (x + m)(x + n). Always pull out any greatest common factor first.
Worked example
Factor x² + 5x + 6.
Find two numbers that multiply to 6 and add to 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3). Check by expanding: x² + 3x + 2x + 6 = x² + 5x + 6.
Factor x² + 7x + 12.
Factor out the greatest common factor: 4x² + 8x.
Mini-lesson
Set the quadratic equal to zero, factor it, then set each factor to zero and solve. The two solutions are the values that make either factor zero.
Worked example
Solve x² - 4x + 3 = 0.
Two numbers with product 3 and sum -4 are -1 and -3: x² - 4x + 3 = (x - 1)(x - 3) = 0. Set each to zero: x - 1 = 0 gives x = 1; x - 3 = 0 gives x = 3.
Solve x² - 6x + 8 = 0.
Solve x² - 9 = 0.
Solve x² + 5x + 6 = 0.
Mini-lesson
When factoring is hard, use x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0. The part under the root, b² - 4ac, is the discriminant: it decides how many real solutions a quadratic has.
Worked example
Solve x² + 2x - 3 = 0.
Here a = 1, b = 2, c = -3. b² - 4ac = 4 + 12 = 16. x = (−2 ± 4) / 2, giving x = 1 and x = -3.
Using the quadratic formula, solve x² + 4x + 1 = 0.
Solve x² - 2x - 2 = 0.
Solve 2x² - 3x - 2 = 0.
Mini-lesson
Vertex form y = a(x - h)² + k has its turning point at (h, k). To complete the square, take half of the x-coefficient, square it, add and subtract, then fold into a square.
Worked example
Write x² + 6x + 8 in vertex form.
Half of 6 is 3; 3² = 9. x² + 6x + 8 = (x² + 6x + 9) + 8 - 9 = (x + 3)² - 1. Vertex: (-3, -1).
For f(x) = x² + 4x + 7, rewritten as f(x) = (x + 2)² + k, what is k?
What is the vertex of f(x) = x² - 8x + 18?
The function f(x) = -(x - 1)² + 6 is in vertex form. What are the coordinates of its maximum?
Mini-lesson
When multiplying like bases, add the exponents: a^m · a^n = a^(m+n). When raising a power to a power, multiply the exponents: (a^m)^n = a^(mn). Negative exponents mean reciprocals, and any nonzero base to the zero power equals 1.
Worked example
Simplify (x³)² · x.
First (x³)² = x⁶ by multiplying the exponents. Then x⁶ · x = x⁷ by adding exponents (the lone x is x¹).
Simplify x⁵ · x³.
Simplify (x²)⁵.
Simplify (3x²)³.
Mini-lesson
The x-intercepts (roots) are where the graph crosses the x-axis, where y = 0. In factored form, each factor set to zero gives a root. A positive leading coefficient opens upward (a minimum); a negative one opens downward (a maximum).
Worked example
Find the x-intercepts of y = (x - 2)(x + 3).
Set each factor to zero: x - 2 = 0 gives x = 2; x + 3 = 0 gives x = -3. The intercepts are (2, 0) and (-3, 0).
What are the x-intercepts of y = (x - 1)(x + 4)?
Which of these quadratic functions opens upward (has a minimum)?
What are the roots of x² - x - 6 = 0?
Mini-lesson
A rational expression is a fraction with a polynomial on top. Factor the numerator (and denominator), then cancel any matching common factors. Cancel only factors, never terms you add or subtract.
Worked example
Simplify (x² - 4) / (x - 2).
Factor the top: x² - 4 = (x - 2)(x + 2). Cancel the (x - 2) in the denominator to get x + 2 (valid when x ≠ 2).
Simplify (x² - 1) / (x + 1).
Simplify 3x⁶ / x².
Mini-lesson
f(3) means replace every x with 3 and evaluate. Two expressions are equivalent if they simplify to the same thing. Use the distributive law, then combine like terms.
Worked example
Given f(x) = 3x + 2, find f(4).
Replace x with 4: f(4) = 3(4) + 2 = 12 + 2 = 14.
If f(x) = x² - 1, what is f(3)?
If g(x) = 2x + 5 and g(a) = 11, what is the value of a?
Which expression is equivalent to 2(x + 3) + x?