SUMMARY: The chapter on Quadratic Equations in Class 10 Mathematics focuses on understanding and solving quadratic equations using various methods. KEY TOPICS: quadratic equation, standard form, factorization method, completing the square, quadratic formula, nature of roots, discriminant, real and distinct roots, real and equal roots, word problems involving quadratic equations
What is the standard form of a quadratic equation?
Aax^2 + bx + c = 0
Bax^2 + b = 0
Cx^2 + bx + c = 0
Dax^2 + bx = 0
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Correct answer: Option 1 — ax^2 + bx + c = 0
Q21 Mark
If the discriminant of a quadratic equation is zero, what can be said about the roots?
AThe roots are real and distinct.
BThe roots are real and equal.
CThe roots are complex.
DThe roots are imaginary.
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Correct answer: Option 2 — The roots are real and equal.
Q31 Mark
Which method can be used to solve the quadratic equation x^2 - 5x + 6 = 0?
AGraphical method only
BFactorization method only
CCompleting the square only
DAll of the above methods
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Correct answer: Option 4 — All of the above methods
Q41 Mark
For the quadratic equation 2x^2 - 4x + 2 = 0, what is the nature of the roots?
AReal and distinct
BReal and equal
CComplex
DImaginary
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Correct answer: Option 2 — Real and equal
Q51 Mark
A rectangular garden has a length that is 2 meters more than its width. If the area of the garden is 48 square meters, what is the width of the garden?
A4 meters
B6 meters
C8 meters
D10 meters
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Correct answer: Option 2 — 6 meters
Short Answer Questions5 questions
Q63 Marks
What is the standard form of a quadratic equation?
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The standard form of a quadratic equation is given by ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
Q73 Marks
Explain the factorization method for solving quadratic equations.
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The factorization method involves expressing the quadratic equation in the form (px + q)(rx + s) = 0, where the product of the factors equals zero. By setting each factor to zero, we can find the values of x that satisfy the equation.
Q83 Marks
What is the quadratic formula and when is it used?
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The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It is used to find the roots of a quadratic equation when it cannot be easily factored.
Q93 Marks
Define the discriminant of a quadratic equation and explain its significance.
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The discriminant of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. It determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there are two equal real roots; and if D < 0, there are no real roots.
Q103 Marks
Solve the quadratic equation x² - 5x + 6 = 0 using the completing the square method.
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To complete the square, rewrite the equation as x² - 5x = -6. Then, add (5/2)² = 6.25 to both sides to get (x - 2.5)² = 0.25. Taking the square root gives x - 2.5 = ±0.5, leading to solutions x = 3 and x = 2.
Long Answer Questions5 questions
Q116 Marks
Solve the quadratic equation 2x^2 - 4x - 6 = 0 using the factorization method. Show all steps involved in the factorization process and verify your solution by substituting the roots back into the original equation.
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To solve the equation 2x^2 - 4x - 6 = 0, first divide the entire equation by 2 to simplify it: x^2 - 2x - 3 = 0. Next, we need to factor the quadratic. We look for two numbers that multiply to -3 (the constant term) and add to -2 (the coefficient of x). The numbers -3 and 1 satisfy these conditions. Thus, we can factor the equation as (x - 3)(x + 1) = 0. Setting each factor to zero gives us the roots x = 3 and x = -1. To verify, substitute these values back into the original equation: for x = 3, 2(3)^2 - 4(3) - 6 = 0, and for x = -1, 2(-1)^2 - 4(-1) - 6 = 0. Both roots satisfy the equation, confirming our solution.
Q126 Marks
Explain the method of completing the square for solving the quadratic equation x^2 + 6x + 5 = 0. Provide a detailed step-by-step explanation and find the roots of the equation.
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To solve the equation x^2 + 6x + 5 = 0 by completing the square, we start by moving the constant term to the other side: x^2 + 6x = -5. Next, we take half of the coefficient of x (which is 6), square it (3^2 = 9), and add it to both sides: x^2 + 6x + 9 = 4. This allows us to rewrite the left side as a perfect square: (x + 3)^2 = 4. Taking the square root of both sides gives us x + 3 = ±2. Solving for x, we find x = -1 and x = -5. Thus, the roots of the equation are -1 and -5.
Q136 Marks
Using the quadratic formula, solve the equation 3x^2 + 2x - 1 = 0. Clearly state the quadratic formula and show all calculations leading to the final answer.
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The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a). For the equation 3x^2 + 2x - 1 = 0, we identify a = 3, b = 2, and c = -1. First, we calculate the discriminant: b² - 4ac = (2)² - 4(3)(-1) = 4 + 12 = 16. Since the discriminant is positive, we have two distinct real roots. Now, substituting into the quadratic formula: x = (-2 ± √16) / (2 * 3) = (-2 ± 4) / 6. This gives us two solutions: x = (2) / 6 = 1/3 and x = (-6) / 6 = -1. Therefore, the roots are x = 1/3 and x = -1.
Q146 Marks
Discuss the nature of roots of the quadratic equation 4x^2 - 12x + 9 = 0. Calculate the discriminant and explain what the result indicates about the roots.
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To determine the nature of the roots of the quadratic equation 4x^2 - 12x + 9 = 0, we first calculate the discriminant using the formula D = b² - 4ac. Here, a = 4, b = -12, and c = 9. Thus, D = (-12)² - 4(4)(9) = 144 - 144 = 0. Since the discriminant is zero, this indicates that the equation has exactly one real root, which means the roots are real and equal. This can also be confirmed by using the quadratic formula: x = (-b ± √D) / (2a) = (12 ± 0) / 8 = 1.5. Therefore, the root is x = 1.5, which is a repeated root.
Q156 Marks
A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 70 square meters, formulate a quadratic equation to represent this situation and solve for the dimensions of the garden.
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Let the width of the garden be x meters. Then the length is x + 3 meters. The area of the rectangle is given by length × width, which leads to the equation: x(x + 3) = 70. Expanding this gives us x² + 3x - 70 = 0. To solve this quadratic equation, we can factor it: (x + 10)(x - 7) = 0. Setting each factor to zero gives us x + 10 = 0 or x - 7 = 0, leading to x = -10 (not feasible) or x = 7. Thus, the width of the garden is 7 meters, and the length is 7 + 3 = 10 meters. Therefore, the dimensions of the garden are 7 meters by 10 meters.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The standard form of a quadratic equation is ax^2 + bx + c = 0.
Reason (R): In this form, a, b, and c are real numbers with a ≠ 0.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The roots of the quadratic equation x^2 - 4x + 4 = 0 are real and distinct.
Reason (R): The discriminant of this equation is zero.
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Correct answer: Option 3 —
A is true, but R is false.
Q181 Mark
Assertion (A): Completing the square can be used to derive the quadratic formula.
Reason (R): The quadratic formula is derived from the process of completing the square.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): If the discriminant of a quadratic equation is negative, the roots are real and equal.
Reason (R): A negative discriminant indicates that the roots are complex.
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Correct answer: Option 4 —
A is false, but R is true.
Q201 Mark
Assertion (A): The factorization method can always be used to solve any quadratic equation.
Reason (R): Some quadratic equations cannot be factored easily or at all.
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Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The standard form of a quadratic equation is ax² + bx + c = 0 where a ≠ 0.
Statement 2: The discriminant of a quadratic equation is given by the formula b² - 4ac.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: A quadratic equation can have at most two real roots.
Statement 2: If the discriminant is negative, the quadratic equation has two distinct real roots.
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Correct answer: Option 4 —
Both statements are false.
Q231 Mark
Statement 1: The roots of the quadratic equation x² - 4x + 4 = 0 are real and equal.
Statement 2: Completing the square can be used to derive the quadratic formula.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The factorization method can be used to solve any quadratic equation.
Statement 2: If the discriminant is zero, the roots of the quadratic equation are real and distinct.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q251 Mark
Statement 1: The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.
Statement 2: A quadratic equation can represent a real-world problem, such as projectile motion.
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Correct answer: Option 1 —
Both statements are true.
