GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra Flashcards

24 sample cards from the Quantitative Reasoning: Algebra deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What condition on the discriminant means a quadratic has exactly one (repeated) real solution?

   b^2 - 4ac = 0, which corresponds to a perfect-square trinomial / the parabola being tangent to the x-axis.

2. Describe the steps to solve a quadratic by completing the square for x^2 + bx + c = 0.

   Move c to the other side, add (b/2)^2 to both sides to form a perfect square, write the left side as (x + b/2)^2, then take the square root of both sides and solve.

3. For x^2 + bx, what term completes the square?

   Add (b/2)^2. Then x^2 + bx + (b/2)^2 = (x + b/2)^2.

4. For ax^2 + bx + c = 0, what are the sum and product of the roots?

   Sum of roots = -b/a; Product of roots = c/a.

5. If the roots of a quadratic are r and s, how can you write the quadratic (leading coefficient 1)?

   x^2 - (r + s)x + rs = 0, i.e., x^2 - (sum)x + (product) = 0.

6. How do you solve an equation reducible to quadratic form, such as x^4 - 5x^2 + 4 = 0?

   Substitute u = x^2 (so x^4 = u^2), giving u^2 - 5u + 4 = 0; solve for u, then back-substitute and solve x^2 = u for x. Here u = 1 or 4, so x = ±1, ±2.

7. What substitution helps solve an equation like x - 5√x + 6 = 0?

   Let u = √x (so x = u^2), giving u^2 - 5u + 6 = 0; solve for u, then x = u^2. Check that u ≥ 0 since √x cannot be negative.

8. What does the function notation f(x) mean, and how do you evaluate f(3)?

   f(x) is the output (value) of function f for input x. To find f(3), substitute 3 for every x in the rule. E.g., if f(x)=2x+1, then f(3)=7.

9. How do you evaluate a composite function f(g(x))?

   Work from the inside out: first compute g(x), then use that result as the input to f. E.g., f(g(2)) means find g(2) first, then plug it into f.

10. In a "symbolism" or made-up-operator problem, how do you handle a defined operation like a ◊ b = a^2 - b?

    Treat the symbol as a rule/recipe: substitute the given values into the definition exactly. E.g., 3 ◊ 5 = 3^2 - 5 = 9 - 5 = 4.

11. What is an arithmetic sequence, and what is its common difference?

    A sequence where each term is obtained by adding a fixed number (the common difference, d) to the previous term. E.g., 2, 5, 8, 11 has d = 3.

12. State the formula for the nth term of an arithmetic sequence.

    a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.

13. State the formula for the sum of the first n terms of an arithmetic sequence.

    S_n = n/2 · (a_1 + a_n) = n/2 · [2a_1 + (n - 1)d].

14. What is a geometric sequence, and what is its common ratio?

    A sequence where each term is obtained by multiplying the previous term by a fixed number (the common ratio, r). E.g., 3, 6, 12, 24 has r = 2.

15. State the formula for the nth term of a geometric sequence.

    a_n = a_1 · r^(n-1), where a_1 is the first term and r is the common ratio.

16. How do you rearrange a formula to solve for a specific variable (e.g., solve A = lw for w)?

    Treat the target variable as the unknown and isolate it using inverse operations on both sides. For A = lw, divide both sides by l to get w = A/l.

17. How are points plotted on the x-y (Cartesian) coordinate plane?

    A point is written as an ordered pair (x, y): move x units horizontally from the origin (right if positive, left if negative), then y units vertically (up if positive, down if negative).

18. Name the four quadrants of the coordinate plane by the signs of (x, y).

    Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), Quadrant IV (+,-), numbered counterclockwise starting from the upper right.

19. State the slope formula between two points (x1, y1) and (x2, y2).

    slope m = (y2 - y1) / (x2 - x1), i.e., rise over run (change in y divided by change in x).

20. What are the slopes of horizontal lines, vertical lines, parallel lines, and perpendicular lines?

    Horizontal line: slope 0. Vertical line: slope undefined. Parallel lines: equal slopes. Perpendicular lines: slopes are negative reciprocals (m1 · m2 = -1).

21. State the slope-intercept form of a line and identify its parts.

    y = mx + b, where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis).

22. State the point-slope form of a line.

    y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line.

23. State the distance formula and the midpoint formula for points (x1, y1) and (x2, y2).

    Distance = √[(x2 - x1)^2 + (y2 - y1)^2]. Midpoint = ((x1 + x2)/2, (y1 + y2)/2).

24. How do you find the x-intercept and y-intercept of a line, and how do you find where two lines intersect?

    x-intercept: set y = 0 and solve for x; y-intercept: set x = 0 and solve for y. Intersection of two lines: solve the two equations simultaneously (substitution or elimination).

What this deck covers

Quantitative Reasoning: Algebra flashcards FAQ