Master Quadratic Equations and Linear Systems: 116 Flashcards from the Discriminant to LU Decomposition
Study the 116-card MB8264 deck: discriminant, completing the square, Vieta's formulas, Gaussian elimination, rank and null space, plus two cards to fix first.
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Most algebra decks stop at the quadratic formula. MB8264 — Quadratic Equations & Systems of Equations (116 cards) does not: it is really two units bolted together. Cards 1–62 drill quadratics, cards 63–116 move to systems of linear equations, and the last thirty quietly become a numerical linear algebra course — LU decomposition, Cholesky, Jacobi iteration, the Moore-Penrose inverse. Spotting that split is the whole trick to studying it efficiently.
Study this deck: Quadratic Equations & Systems of Equations
1 / 10Quadratic equation
Tap the card for the answer · swipe right if you knew it, left if not
Eight cards that show the range
| Front | Back |
|---|---|
| Discriminant | The expression b² − 4ac under the square root in the quadratic formula |
| The discriminant is negative | Two distinct complex conjugate roots (no real roots) |
| What do you add to x² + bx to complete the square? | (b/2)², the square of half the coefficient of x |
| Vieta's formulas for a quadratic equation | x₁ + x₂ = −b/a and x₁·x₂ = c/a |
| X-coordinate of the vertex | x = −b/(2a) |
| Overdetermined system | More equations than unknowns, generally no solution |
| Rouché–Capelli theorem: when is a system inconsistent? | When the rank of the augmented matrix exceeds the rank of the coefficient matrix |
| Null space of a matrix | The solution set to the homogeneous system Ax = 0 |
Everything the deck covers
Quadratic half: standard form, reduced (monic) form x² + px + q = 0, double roots, factoring by inspection, the Zero Factor Property, completing the square, vertex form y = a(x − h)² + k, axis of symmetry, x- and y-intercepts, parabola orientation, quadratic factorization a(x − r)(x − s), the golden ratio as a root of x² − x − 1 = 0, Lill's method, Carlyle circles, and applications in projectile motion and weak-acid pH.
Linear systems half: consistent vs inconsistent, underdetermined vs overdetermined, linear independence, hyperplanes in n dimensions, augmented matrices, the three elementary row operations, Gaussian and Gauss–Jordan elimination, reduced row echelon form, rank, Cramer's rule, pivoting, sparse matrices, homogeneous systems, trivial solutions, free vs dependent variables, and linearization.
Two fixes to make before you grind it
Restore the missing question words. Several fronts were clipped during authoring: "Vieta's formulas used for", "standard form → vertex form", "Brahmagupta's solution equivalent to", "Equations to be linearly independent". Read them as fill-in-the-blank prompts ("What are Vieta's formulas used for?") or rewrite the fronts — otherwise you will blank on cards you actually know.
Watch the sign convention on two formula cards. Brahmagupta's root is given as x = (√(4ac + b²) − b)/(2a), which only matches the deck's own standard form if you write the equation as ax² + bx = c, not ax² + bx + c = 0. And Muller's alternative, x = −2c/(b ± √(b² − 4ac)), is not trivia: it is the form you use when b² is much larger than 4ac, exactly the catastrophic-cancellation case the deck's "loss of significance" card warns about.
How to sequence it
Run cards 1–62 until the discriminant triggers the root-count answer instantly, then treat 63–116 as a separate deck — the vocabulary there (rank, null space, echelon form) is linear algebra, not Algebra II, and mixing the two early just slows both down.
Find MB8264 in your library on Flip It. For the next step up, see our study guide to exponents, logarithms, complex numbers and matrices.
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