NCERT Class 10 Maths — Chapter 4

Exercise 4.4: Nature of Roots

Use the discriminant (b² - 4ac) to determine whether a quadratic has two distinct real roots, one repeated root, or no real roots — without solving.

discriminantnature of rootsreal and equal rootsno real roots

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Extra Practice Questions

These questions cover the same concepts as Exercise 4.4. Try solving them to build confidence before or after the textbook exercise.

The product of two consecutive positive integers is 306. Which of the following quadratic equations represents this situation?

A.x² + x + 306 = 0

B.x² - x + 306 = 0

C.x² + x - 306 = 0

D.x² - x - 306 = 0

Consider the quadratic equation x² - 2(a + b)x + (a² + b² + 2c²) = 0. If its roots are real and equal, then which of the following statements must be true?

A.a = b

B.a = b = c

C.ab = c²

D.a = b and c = 0

Find the real roots of the equation: x + (1/x) = 5/2, where x ≠ 0.

A.x = 1/2, 2

B.x = 1, 2

C.x = -1/2, -2

D.x = 1/2, -2

Consider the statement: 'If the quadratic equation ax² + bx + c = 0 has rational roots, then b² - 4ac must be a perfect square, even if a, b, c are not rational.' Is this statement always true, sometimes true, or always false?

A.Always true

B.Sometimes true

C.Always false

D.Cannot be determined

Find the roots of the quadratic equation 6x² + 17x + 5 = 0 by factorization.

A.x = -1/3, x = -5/2

B.x = 1/3, x = 5/2

C.x = -1/2, x = -5/3

D.x = 1/2, x = 5/3

Which of the following statements is true about the quadratic equation x² + 2x + 5 = 0?

A.It has two distinct real roots.

B.It has two equal real roots.

C.It has no real roots.

D.It has one real root and one imaginary root.

The sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.

A.Sides are 12 m and 18 m

B.Sides are 15 m and 21 m

C.Sides are 10 m and 16 m

D.Sides are 9 m and 15 m

If the roots of the quadratic equation ax² + bx + c = 0 are rational, which of the following statements about its discriminant (D = b² - 4ac) must be true?

A.D must be positive.

B.D must be a perfect square (and non-negative).

C.D must be zero.

D.D can be any real number.

Using the quadratic formula, find the roots of the equation √2 x² + 7x + 5√2 = 0.

A.x = -5/√2, x = -√2

B.x = 5/√2, x = √2

C.x = -5√2, x = -√2

D.x = 5√2, x = √2

Ravi is trying to solve the quadratic equation x² - 7x + 12 = 0 by factorization. He writes the first step as x² - 3x - 4x + 12 = 0. Which of the following statements about Ravi's first step is true?

A.Ravi has made a mistake in splitting the middle term.

B.Ravi has correctly split the middle term, and the next step is x(x-3) - 4(x-3) = 0.

C.Ravi should have split the middle term as -2x - 5x.

D.Ravi should have used the quadratic formula instead of factorization.

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