Quadratic Equations · Class 10 Mathematics: 60 flashcards
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1. What is the standard form of a quadratic equation?
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$ax^2 + bx + c = 0$ with a, b, c real and $a \ne 0$.
2. Why must $a \ne 0$?
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With $a = 0$ the $x^2$ term disappears and the equation becomes linear.
3. May b or c be zero?
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Yes — $x^2 - 9 = 0$ and $3x^2 + 5x = 0$ are both quadratic.
4. How do you decide whether an equation is quadratic?
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Expand and simplify it first, then check that the highest power of x is 2.
5. Is $(x - 2)(x + 1) = (x - 1)(x + 3)$ quadratic?
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No — the $x^2$ terms cancel, leaving the linear equation $3x = 1$.
6. In $x^2 - 5x + 6 = 0$, what is b?
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$b = -5$ — the sign is part of the coefficient.
7. What makes a number $\alpha$ a root?
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Substituting it gives $a\alpha^2 + b\alpha + c = 0$.
8. Is 1 a root of $2x^2 - 3x + 1 = 0$?
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Yes: $2 - 3 + 1 = 0$.
9. How are roots and zeroes related?
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The roots of $ax^2 + bx + c = 0$ are the zeroes of the polynomial $ax^2 + bx + c$ — the same numbers.
10. At most how many roots can a quadratic equation have?
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Two.
11. Write two consecutive positive integers algebraically.
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$x$ and $x + 1$.
12. Write two consecutive even integers algebraically.
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$x$ and $x + 2$.
13. A rectangle's length is one more than twice its breadth and its area is 528. Form the equation.
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$x(2x + 1) = 528$, i.e. $2x^2 + x - 528 = 0$.
14. The product of two consecutive positive integers is 306. Form the equation.
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$x^2 + x - 306 = 0$.
15. Rohan's mother is 26 years older. In 3 years the product of their ages is 360. Form the equation.
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$(x + 3)(x + 29) = 360$, i.e. $x^2 + 32x - 273 = 0$.
16. What is the formula linking distance, speed and time?
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$\text{time} = \frac{\text{distance}}{\text{speed}}$.
17. A train covers 480 km; 8 km/h slower would take 3 hours more. Form the equation.
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$\frac{480}{x-8} - \frac{480}{x} = 3$, which simplifies to $x^2 - 8x - 1280 = 0$.
18. Is $(x + 2)^3 = 2x(x^2 - 1)$ quadratic?
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No — after expanding, the highest power is 3, so it is cubic.
19. What must a 'represent the situation' answer include to earn full marks?
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The letter defined with its unit, the equation formed, and the equation simplified to standard form.
20. In $3x^2 - 7 = 0$, what are a, b and c?
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$a = 3$, $b = 0$, $c = -7$.
21. State the zero-product rule.
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If $pq = 0$ then $p = 0$ or $q = 0$.
22. Why does factorisation solve a quadratic equation?
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It turns the equation into a product equal to zero, so each factor gives a linear equation.
23. What must the two numbers in the split satisfy?
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They multiply to $ac$ and add to $b$.
24. For $2x^2 - 5x + 3$, what is the split?
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$-2$ and $-3$, since $ac = 6$ and $b = -5$.
25. For $6x^2 - x - 2$, what is the split?
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$-4$ and 3, since $ac = -12$ and $b = -1$.
26. How do you know the split was correct?
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After grouping, the bracket left over is identical in both pairs.
27. The factor $(x - 2)$ gives which root?
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$x = 2$ — set the bracket to zero and solve.
28. Solve $x^2 - 5x + 6 = 0$.
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$(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.
29. Solve $2x^2 - 5x + 3 = 0$.
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$(x-1)(2x-3) = 0$, so $x = 1$ or $x = \frac32$.
30. Solve $6x^2 - x - 2 = 0$.
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$(3x-2)(2x+1) = 0$, so $x = \frac23$ or $x = -\frac12$.
31. What should you do before hunting for a split?
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Divide the whole equation by any factor common to all three terms.
32. How many roots does $(x + 2)^2 = 0$ have?
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Two equal roots, both $-2$.
33. Can you write $x - 1 = 6$ from $(x-1)(x-2) = 6$?
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No — the zero-product rule only applies when the product is 0.
34. Solve $x^2 + x - 306 = 0$.
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$(x+18)(x-17) = 0$, so $x = 17$ or $x = -18$.
35. When must a root be rejected?
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When the quantity it represents cannot take that value — a negative length, speed, age or count.
36. Solve $x^2 - 8x - 1280 = 0$ and give the train's speed.
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$(x-40)(x+32) = 0$; a speed cannot be negative, so it is 40 km/h.
37. What is the first step of the method?
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Bring the equation to standard form $ax^2 + bx + c = 0$.
38. Where should you verify a root?
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In the ORIGINAL equation.
39. What does it mean if no integer split exists?
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Factorisation is the wrong tool; use the quadratic formula.
40. Solve $x^2 - 7x - 60 = 0$.
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$(x-12)(x+5) = 0$, so $x = 12$ or $x = -5$.
41. What is the discriminant of $ax^2 + bx + c = 0$?
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$D = b^2 - 4ac$.
42. What does $D > 0$ mean?
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Two distinct real roots.
43. What does $D = 0$ mean?
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Two equal real roots, both $-\frac{b}{2a}$.
44. What does $D < 0$ mean?
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No real roots.
45. State the quadratic formula.
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$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
46. When is the quadratic formula valid?
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When $b^2 - 4ac \ge 0$.
47. If $b = -5$, what is $b^2$?
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25 — squaring always gives a non-negative result.
48. If $b = -5$, what is $-b$ in the formula?
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$+5$.
49. For $2x^2 + x - 3 = 0$, what is $-4ac$?
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$-4(2)(-3) = +24$.
50. Find D for $3x^2 - 5x + 2 = 0$.
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$25 - 24 = 1$.
51. Solve $3x^2 - 5x + 2 = 0$ by the formula.
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$x = \frac{5 \pm 1}{6}$, giving 1 and $\frac23$.
52. Find D for $x^2 + x - 1 = 0$ and say what it means.
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$D = 5$ — positive but not a perfect square, so the roots are real and irrational.
53. Solve $x^2 + x - 1 = 0$.
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$x = \frac{-1 \pm \sqrt5}{2}$.
54. When are the roots rational?
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When $D$ is a positive perfect square (or zero).
55. What condition gives equal roots?
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$D = 0$, i.e. $b^2 = 4ac$.
56. For what k does $2x^2 + kx + 3 = 0$ have equal roots?
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$k^2 = 24$, so $k = \pm 2\sqrt6$.
57. For what k does $kx^2 - 2kx + 6 = 0$ have equal roots?
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$k = 6$; $k = 0$ is rejected because it leaves no $x^2$ term.
58. Can a rectangular park of perimeter 80 m have area 500 m²?
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No — the equation $x^2 - 40x + 500 = 0$ has $D = -400 < 0$.
59. Can a rectangular park of perimeter 80 m have area 400 m²?
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Yes — $D = 0$, giving a 20 m by 20 m square.
60. What three things should a 'nature of roots' answer contain?
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The values of a, b and c; the computed D with working; the conclusion in words.
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