Quadratic Equations: 25 flashcards
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1. What is the standard form of a quadratic equation?
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$ax^2 + bx + c = 0$, where $a \neq 0$.
2. Why must $a \neq 0$ in a quadratic equation?
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If $a = 0$, the $x^2$ term disappears and the equation becomes linear, not quadratic.
3. State the quadratic formula.
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$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
4. What is the discriminant of a quadratic equation?
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$D = b^2 - 4ac$
5. If $D > 0$, what is the nature of the roots?
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Two distinct real roots.
6. If $D = 0$, what is the nature of the roots?
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Two equal real roots (one repeated root).
7. If $D < 0$, what is the nature of the roots?
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No real roots (complex roots).
8. What is the sum of the roots of $ax^2 + bx + c = 0$?
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$\alpha + \beta = -\dfrac{b}{a}$
9. What is the product of the roots of $ax^2 + bx + c = 0$?
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$\alpha\beta = \dfrac{c}{a}$
10. What is the zero-product rule?
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If $AB = 0$, then $A = 0$ or $B = 0$.
11. What is the factored form of a quadratic and what does it reveal?
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$(x - p)(x - q) = 0$; the roots are $p$ and $q$ immediately.
12. What is the vertex form of a quadratic and where is the vertex?
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$y = a(x - h)^2 + k$; the vertex is at $(h, k)$.
13. At what $x$-value is the vertex of $y = ax^2 + bx + c$ located?
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$x = -\dfrac{b}{2a}$
14. What shape is the graph of a quadratic equation?
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A parabola — a symmetric U-shaped curve.
15. How does the sign of $a$ affect the parabola?
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If $a > 0$, it opens upward; if $a < 0$, it opens downward.
16. What is the relationship between the roots of a quadratic and its graph?
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The roots are the $x$-intercepts of the parabola.
17. Which method is fastest when roots are simple integers?
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Factorisation.
18. Which method always works for any quadratic?
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The quadratic formula.
19. What is completing the square used for?
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It rewrites a quadratic as a perfect square plus a constant, and it is the derivation behind the quadratic formula.
20. What common mistake do students make with the $\pm$ in the quadratic formula?
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They compute only one root instead of both. The $\pm$ means you must calculate two values.
21. Name two real-world situations modelled by quadratic equations.
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Projectile motion (height of a thrown ball) and area problems (e.g., garden dimensions).
22. What does the vertex represent in an optimisation problem?
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The maximum or minimum value — e.g., maximum profit or minimum cost.
23. How do you check if your factorised roots are correct?
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Verify that their sum equals $-b/a$ and their product equals $c/a$.
24. What is the Latin origin of the word 'quadratic' and what does it mean?
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From Latin *quadratus*, meaning 'square'.
25. If a quadratic has roots 3 and $-2$, what is the equation?
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$x^2 - x - 6 = 0$ (since sum $= 1$ and product $= -6$).
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