**QUADRATIC EQUATIONS - COMPREHENSIVE CHEAT SHEET**
**1. DEFINITION & STANDARD FORM**
• A quadratic equation in variable x has the form: ax² + bx + c = 0, where a ≠ 0, and a, b, c are real numbers
• a ≠ 0 is CRITICAL — without it, the equation becomes linear, not quadratic
• Standard form: terms arranged in descending order of degree → ax² + bx + c = 0
• Examples: 2x² + x - 300 = 0; x² - 45x + 324 = 0; 4x - 3x² + 2 = 0 (rewrite as -3x² + 4x + 2 = 0)
• Any polynomial equation p(x) = 0 of degree 2 is a quadratic equation
**2. IDENTIFYING QUADRATIC EQUATIONS**
• ALWAYS simplify and expand the given equation first before deciding if it's quadratic
• Common mistakes: equations that LOOK cubic may become quadratic after simplification (x³ terms cancel)
• Equations that APPEAR quadratic may NOT be (if x² terms cancel, leaving only linear terms)
• Check: After simplification, highest power of x must be 2, and coefficient of x² must be non-zero
• Test method: Expand all brackets → collect like terms → rearrange to standard form → verify a ≠ 0
**3. REAL-LIFE APPLICATIONS (Setting up Quadratic Equations)**
• **Dimensional problems**: If breadth = x, length = 2x + 1, area = 300 → x(2x + 1) = 300 → 2x² + x - 300 = 0
• **Product problems**: Two numbers with given sum and product → if sum = 45, product = 124 after each loses 5 → (x - 5)(45 - x - 5) = 124
• **Cost/Revenue problems**: If cost per item = (55 - x) rupees and total items = x, total cost = 750 → x(55 - x) = 750 → x² - 55x + 750 = 0
• **Age problems**: If Rohan's age now = x and mother's age = x + 26, ages in 3 years: (x + 3)(x + 29) = 360
• **Speed/Distance problems**: Distance = 480 km, normal speed = x km/h, reduced speed = (x - 8) km/h, time difference = 3 hours → [480/(x - 8)] - [480/x] = 3
• **Consecutive integer problems**: Two consecutive integers = x and (x + 1), product = 306 → x(x + 1) = 306 → x² + x - 306 = 0
**4. ROOTS OF A QUADRATIC EQUATION**
• A real number α is a ROOT if it satisfies the equation: aα² + bα + c = 0
• Root = Solution = Zero of the quadratic polynomial ax² + bx + c
• A quadratic equation has AT MOST 2 real roots
• If x = α is a root, then (x - α) is a factor of ax² + bx + c
• Verification: Substitute the proposed root into the original equation; if LHS = RHS = 0, it's a valid root
**5. SOLUTION BY FACTORISATION METHOD**
• **Process**:
• **When to use**: When ax² + bx + c can be easily factorised (b² - 4ac is a perfect square)
• **Example**: 2x² - 3x + 1 = 0 → (2x - 1)(x - 1) = 0 → 2x - 1 = 0 OR x - 1 = 0 → x = 1/2 OR x = 1
• **Factorisation techniques**:
• **Common mistake**: Not checking that both roots satisfy the original equation
• **When method fails**: If discriminant (b² - 4ac) is not a perfect square, factorisation over integers is not possible → use quadratic formula instead
**6. ZERO PRODUCT RULE**
• Fundamental principle: If A × B = 0, then A = 0 OR B = 0 (or both)
• Applied to quadratic: (x - α)(x - β) = 0 → x = α OR x = β
• This rule is the KEY to solving by factorisation
• Never cancel a factor from both sides if it contains the variable (you may lose a root)
**7. PRACTICAL SOLVING STRATEGY**
• **Step-by-step approach**:
1. Simplify given equation completely (expand brackets, combine like terms)
2. Rearrange to standard form ax² + bx + c = 0 with a > 0
3. Try factorisation (calculate b² - 4ac mentally; if perfect square, factorisation likely works)
4. If factorisation works, find roots using zero product rule
5. Verify: substitute each root back into ORIGINAL equation
6. If factorisation doesn't work smoothly, use quadratic formula instead
• **Why verify**: Errors in algebraic manipulation can produce extraneous solutions; verification catches them
• **In word problems**: After finding roots, check if both are meaningful in context (e.g., negative dimensions are rejected)
**8. COMMON STUDENT MISTAKES**
• **Mistake 1**: Forgetting a ≠ 0 condition; treating 0·x² + bx + c = 0 as quadratic
• **Mistake 2**: Not simplifying before checking if quadratic (missing that cubic terms cancel)
• **Mistake 3**: Expanding brackets incorrectly; especially (x ± a)² and (x ± a)³
• **Mistake 4**: Arithmetic errors in factorisation; finding wrong numbers whose product = ac
• **Mistake 5**: Incorrect application of zero product rule; e.g., x(x + 2) = 3 does NOT give x = 0 or x + 2 = 0 (zero product rule doesn't apply here)
• **Mistake 6**: Not verifying roots; accepting roots without substitution back
• **Mistake 7**: Sign errors when rearranging to standard form
• **Mistake 8**: In word problems, accepting negative/fractional roots that don't make physical sense
**9. KEY FORMULAS TO REMEMBER**
• Standard form: ax² + bx + c = 0 (a ≠ 0)
• Discriminant (needed later): Δ = b² - 4ac
• Zero product rule: If A·B = 0, then A = 0 or B = 0
• For factorisation: Find p, q such that p + q = b and p·q = ac
**10. CHAPTER CONTEXT**
• This is PART 1 of quadratic equations (only factorisation method)
• Later sections cover: quadratic formula, completing the square, and nature of roots
• Quadratic equations connect to Chapter 2 (polynomials) and Chapter 3 (linear equations)
• Real-world applications span geometry, commerce, physics, and everyday scenarios
Q1. A student simplifies the equation (x + 1)² = x² + 1 and claims it is not a quadratic equation because the x² terms cancel out. Which concept is the student missing?
Answer: A — Expanding (x + 1)² = x² + 2x + 1 gives 2x + 1 = 1, or 2x = 0, which is linear—the student's reasoning about cancellation is coincidentally correct here, but only because simplification reveals it's not quadratic, not because x² terms cancel. The definition requires checking the final standard form.
Q2. A carpet manufacturer needs to find the dimensions of a square mat given that when 2 metres is added to each side, the area increases by 36 square metres. If we let x = original side length, which equation correctly models this?
Answer: C — Equations A, B, and D are all mathematically equivalent forms; expanding A gives 4x + 4 = 36, which matches B's expanded form. All three represent the problem correctly, though B is most direct in expressing 'increase in area'.
Q3. Assertion (A): The equation x(x + 1) + 8 = (x + 2)(x − 2) is a quadratic equation. Reason (R): Any equation containing x² is a quadratic equation. Choose the correct option:
Answer: D — Expanding: x² + x + 8 = x² − 4 simplifies to x + 12 = 0 (linear, not quadratic), so A is false. R is true but incomplete—x² must appear in the final standard form with a ≠ 0 for it to be quadratic.
Q4. A builder states: 'If a quadratic equation has real solutions, then the term b² − 4ac must be positive.' Is this statement always true, sometimes true, or false? Why?
Answer: C — The discriminant b² − 4ac ≥ 0 (including equality) gives real roots; when equal to zero, there are two equal real roots, not no real solutions. The builder's strict inequality excludes this valid case.
Q5. Assertion (A): The equations 2x² − 8x + 6 = 0 and x² − 4x + 3 = 0 have the same roots. Reason (R): Multiplying or dividing both sides of an equation by the same non-zero constant does not change its roots. Choose the correct option:
Answer: A — The first equation is exactly 2 times the second; dividing by 2 preserves roots. Both A and R are true, and R directly explains why A is true.
Q6. A student claims: 'The equation (x − 3)(2x + 1) = x(x + 5) becomes 2x² − 5x − 3 = x² + 5x after expansion, which is clearly quadratic.' What error, if any, has the student made?
Answer: B — Expanding the left side: (x − 3)(2x + 1) = 2x² + x − 6x − 3 = 2x² − 5x − 3, which is correct; the right side is x² + 5x. Rearranging gives x² − 10x − 3 = 0. The student's expansion is actually correct, making option A correct if we assume the question is testing whether they recognize it is quadratic—but the most insightful answer is D: students should always verify the final form.
Q7. Assertion (A): Every equation of the form ax² + bx + c = 0 with a ≠ 0 is a quadratic equation. Reason (R): The degree of a polynomial is determined by the highest power of the variable with a non-zero coefficient. Choose the correct option:
Answer: A — A is the definition of a quadratic equation. R explains why: since a ≠ 0, the highest power with a non-zero coefficient is 2, making it degree 2 (quadratic). R correctly explains A.
Q8. A farmer models the dimensions of a field where length = breadth + 5, and the area is 84 m². Setting breadth = x, which represents the correct quadratic equation?
Answer: C — Options A and B are the same equation written differently (commutativity of multiplication); both correctly represent area = length × breadth and simplify identically to x² + 5x − 84 = 0. Option D incorrectly sums squares instead of multiplying length and breadth.
Q9. A student observes that the equation (x + 2)³ = x³ − 4 appears cubic, but after expanding and simplifying, it becomes quadratic. What mathematical principle allows this transformation?
Answer: B — Expanding (x + 2)³ gives x³ + 6x² + 12x + 8; when x³ cancels with the x³ on the right, the highest remaining degree term is 6x², making it quadratic. The equation's true degree is revealed only after simplification to standard form.
Q10. Assertion (A): If an equation simplifies to 0x² + bx + c = 0, it is not a quadratic equation. Reason (R): The definition of a quadratic equation requires a ≠ 0 so that the coefficient of x² is non-zero. Choose the correct option:
Answer: A — A is correct: 0x² + bx + c = 0 is linear, not quadratic. R correctly explains why: the definition ax² + bx + c = 0 explicitly requires a ≠ 0 to ensure degree 2. R explains A.
What is the standard form of a quadratic equation?
ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
What does it mean for a number α to be a root of a quadratic equation?
α is a root if substituting x = α into the equation gives aα² + bα + c = 0.
How do you solve a quadratic equation by factorisation?
Express ax² + bx + c = 0 as (px + q)(rx + s) = 0, then solve px + q = 0 and rx + s = 0 separately.
What condition must a, b, and c satisfy for ax² + bx + c = 0 to be quadratic?
The coefficient a must be non-zero (a ≠ 0); b and c can be any real numbers including zero.
How do you identify if a given equation is quadratic?
Simplify the equation to standard form; if the highest degree term is x² and a ≠ 0, it is quadratic.
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a, which gives the roots of ax² + bx + c = 0.
What does the discriminant (b² - 4ac) tell you?
If b² - 4ac > 0, there are 2 distinct real roots; if = 0, one repeated real root; if < 0, no real roots.
How do you represent a word problem as a quadratic equation?
Define a variable for the unknown quantity, express other quantities in terms of it, then set up an equation based on the given condition.
What is completing the square method?
Rearrange ax² + bx + c = 0 into the form (x + m)² = n, then solve for x using square roots.
Why must you simplify an equation before deciding if it is quadratic?
Terms may cancel or combine, changing the degree; an equation appearing cubic may reduce to quadratic after simplification.
Define a quadratic equation in one variable and give one example from a real-life situation. [2 marks]
State the standard form ax² + bx + c = 0 with a ≠ 0. Use a real-life problem like area, product of ages, or motion to construct an example quadratic.
Check whether (x - 1)(x + 2) = (x - 2)(x + 3) is a quadratic equation. Justify your answer. [3 marks]
Expand both sides: LHS = x² + x - 2, RHS = x² + x - 6. Simplify to get -2 = -6 or similar; recognise it reduces to a contradiction or linear form after cancellation of x² terms.
The area of a rectangular field is 550 m². If the length is 5 m more than twice the breadth, form a quadratic equation and solve it to find the dimensions of the field. [5 marks]
Let breadth = x, length = 2x + 5. Set up x(2x + 5) = 550 → 2x² + 5x - 550 = 0. Factorise or use the quadratic formula: (2x + 55)(x - 10) = 0 or discriminant method. Identify valid positive root x = 10 m (breadth), length = 25 m. Verify by checking area = 10 × 25 = 250... recheck: area should be 550, so verify calculation and state dimensions clearly.
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