Quadratic Equations – important questions for Boards 2027
CBSE Class 10 Maths
Which statement correctly describes how a quadratic equation is obtained from a quadratic polynomial ax² + bx + c, where a ≠ 0? (A) Set it equal to 1. (B) Set it equal to zero. (C) Set its derivative equal to zero. (D) Divide it by x.
Answer
(B) Set the polynomial equal to zero — equating the quadratic polynomial to zero gives a quadratic equation.
State the standard form of a quadratic equation in x and the conditions on its coefficients. What is the degree of its polynomial?
Answer
A quadratic equation in x has standard form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Its polynomial has degree 2.
For the equation 2x² − 5x + 3 = 0, state its corresponding quadratic polynomial and explain the relationship between the roots of the equation and the zeroes of that polynomial.
Answer
1. The corresponding quadratic polynomial is 2x² − 5x + 3, the left-hand expression in the equation. A zero is a value that makes this polynomial zero. 2. A real number α is a root of 2x² − 5x + 3 = 0 when 2α² − 5α + 3 = 0. Therefore, the roots of the equation and the zeroes of its corresponding polynomial are the same.
Simplify each equation and determine whether it is quadratic: (i) x(x + 1) + 8 = (x + 2)(x − 2); (ii) (x + 2)³ = x³ − 4. Give the resulting simplified equation in standard form whenever it is quadratic, and explain why simplification is needed before classification.
Answer
Simplifying an equation before classification reveals its actual polynomial degree and whether its quadratic terms remain. 1. For (i), x(x + 1) + 8 = x² + x + 8, while (x + 2)(x − 2) = x² − 4. Subtracting x² from both sides gives x + 12 = 0, a linear equation, so it is not quadratic. 2. For (ii), expansion gives x³ + 6x² + 12x + 8 = x³ − 4. Subtracting x³ and simplifying gives 6x² + 12x + 12 = 0; dividing by 6 gives x² + 2x + 2 = 0. This has standard form ax² + bx + c = 0 with a = 1 ≠ 0, so it is quadratic. 3. These examples show that the appearance of the original expression is not decisive: equal terms can cancel, or expansion can reveal the remaining degree. Classification depends on the simplified equation, which must be reduced to determine its true degree.
Without solving the equations, classify the roots of each equation as two distinct real roots, two equal real roots, or no real roots: (i) x² − 5x + 6 = 0; (ii) x² − 4x + 4 = 0; (iii) x² + x + 1 = 0. Justify each classification using the discriminant.
Answer
The discriminant b² − 4ac determines the nature of the real roots of a quadratic equation. 1. For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive; therefore, the equation has two distinct real roots. 2. For x² − 4x + 4 = 0, a = 1, b = −4 and c = 4. The discriminant is (−4)² − 4(1)(4) = 16 − 16 = 0; therefore, the equation has two equal real roots. 3. For x² + x + 1 = 0, a = 1, b = 1 and c = 1. The discriminant is 1² − 4(1)(1) = 1 − 4 = −3, which is negative; therefore, the equation has no real roots. Thus, the sign of the discriminant classifies each equation without finding its roots.
A park boundary contains a gate AB of length 13 m. A pole P is to be placed between A and B, with AP = x + 7 m and PB = x m, and ∠APB is a right angle. (a) Use Pythagoras’ theorem to form a quadratic equation in x. (b) State the quadratic formula and explain the discriminant condition for real roots. (c) Use the discriminant to decide whether the placement is possible. (d) Find the roots and explain which root can represent the distance x.
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A rectangular hall has breadth x metres and length one metre more than twice its breadth. Its area is 300 m². Which equation represents the area condition? (A) 2x + 1 = 300 (B) x(2x + 1) = 300 (C) x + 2x + 1 = 300 (D) 2x² + 1 = 300
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John and Jivanti together have 45 marbles. After each loses 5 marbles, the product of their remaining amounts is 124. If John initially has x marbles, form a quadratic equation for x in standard form.
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Solve 6x² − x − 2 = 0 by factorisation. Show the linear factors and the equations obtained by setting each factor equal to zero.
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A factory produces x toys in a day, and the production cost per toy is (55 − x) rupees. The total production cost is ₹750. Form a quadratic equation in standard form to represent this situation, showing how the total cost is obtained.
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6 more probable questions in this chapter, every one with a full answer.
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