Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the discriminant formula for a quadratic equation ax2 + bx + c = 0.

Discriminant (Δ) = b2−4acb2+4ac4ac−b2a2−4bc

Perfect! The discriminant determines the nature of roots.

(2) Find the roots of the equation x2 - 16 = 0.

Roots: x = and (Enter in increasing order of value)

Correct! x = 4 and x = -4 are the two roots.

(3) If one root of a quadratic equation is 4 and the other is -1, write the equation.

Equation: x22x2 + + = 0

Excellent! (x - 4)(x + 1) = x2 - 3x - 4 = 0.

(4) What is the nature of roots of the equation x2 - 5x + 7 = 0? imaginary/complexreal and equalreal and distinct as the Δ =

Perfect! Negative discriminant means no real roots.

Short Answer Questions (2 Marks Each)

(1) Solve 2x2 + 7x + 3 = 0 using factorization.

Roots: x = and (Enter in increasing order of value)

Excellent factorization! Always verify by substitution.

(2) Find the discriminant and nature of roots of x2 + 4x + 5 = 0.

Discriminant: Δ =

Since Δ = -4 < 0, nature of roots: No real roots (imaginary)Real and equalReal and distinctRational and unequal

Perfect! Negative discriminant always means imaginary roots.

(3) Solve the equation x2 + 2x - 15 = 0 by quadratic formula.

Roots: x = and

Excellent! x = 3 and x = -5.

(4) For what value of k does the equation x2 + kx + 4 = 0 have equal roots?

Therefore: k = and

Perfect! k = 4 or k = -4 gives equal roots.

Long Answer Questions (4 Marks Each)

(1) A number is such that the square of the number is 5 more than four times the number. Find the number.

The numbers are: and

Excellent! Both solutions are mathematically valid.

(2) The area of a rectangle is 528 cm2. If the length is one more than twice the breadth, find the dimensions of the rectangle.

Standard form: 2x2x2 + + = 0

Since breadth must be positive: x = cm

Length = cm

Perfect! Dimensions are 16 cm × 33 cm.

(3) A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 48 minutes less. Find the speed of the train.

Speed of train = km/h

Excellent problem-solving! The train's speed is 45 km/h.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The value of discriminant of 3x2 - 5x + 2 = 0 is:

(a) 1 (b) -1 (c) 25 (d) 49

Correct! Δ = −52 - 4(3)(2) = 25 - 24 = 1.

(2) The equation x2 + 6x + 9 = 0 has:

(a) No real roots (b) Two distinct roots (c) Equal real roots (d) Two imaginary roots

Correct! This is x+32 = 0, so Δ = 0 means equal roots.

(3) If the roots of x2 + px + 4 = 0 are reciprocals of each other, then:

(a) p = 2 (b) p = 0 (c) p = -2 (d) p = 4

Correct! The value of p is equal to -2.

(4) The quadratic equation with roots 2 and -3 is:

(a) x2 + x - 6 = 0 (b) x2 - x - 6 = 0 (c) x2 - 5x + 6 = 0 (d) x2 + 5x + 6 = 0

Correct! Sum = 2 + (-3) = -1, Product = 2 × (-3) = -6. So x2 - (sum)x + product = x2 + x - 6 = 0.

(5) If the sum of roots is 7 and the product is 10, the equation is:

(a) x2 + 7x + 10 = 0 (b) x2 - 7x + 10 = 0 (c) x2 - 7x - 10 = 0 (d) x2 + 7x - 10 = 0

Correct! Using x2 - (sum)x + product = x2 - 7x + 10 = 0.

(6) The roots of 2x2 + 3x - 5 = 0 are:

(a) Real and equal (b) Imaginary (c) Real and distinct (d) Zero

Correct! Δ = 9 - 4(2)(-5) = 9 + 40 = 49 > 0, so real and distinct.

(7) If the roots of the quadratic equation are equal, the value of discriminant is:

(a) Positive (b) Negative (c) Zero (d) One

Correct! Equal roots occur when Δ = 0.

(8) The quadratic equation whose sum and product of roots are both 4 is:

(a) x2 - 4x + 4 = 0 (b) x2 + 4x + 4 = 0 (c) x2 - 4x - 4 = 0 (d) x2 + 4x - 4 = 0

Correct! Using x2 - (sum)x + product = x2 - 4x + 4 = 0.

(9) If one root of a quadratic equation is 3 and the equation is x2 - 6x + c = 0, the value of c is:

(a) 9 (b) -9 (c) 3 (d) -3

Correct! Substituting x = 3: 9 - 18 + c = 0, so c = 9.

(10) Which of the following quadratic equations has imaginary roots?

(a) x2 + 5x + 6 = 0 (b) x2 - 4x + 4 = 0 (c) x2 + 2x + 3 = 0 (d) x2 - x - 6 = 0

Correct! Δ = 4 - 12 = -8 < 0, so imaginary roots.

Quadratic Equations Challenge

Determine whether these statements about quadratic equations are True or False:

Quadratic Equations Quiz