Quadratic Equations

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b and c are real numbers and a is not zero. Such equations arise from areas of plots, speeds of journeys and ages. A real number alpha is a root of the equation if a(alpha)^2 + b(alpha) + c = 0, so its roots are the zeroes of the polynomial ax^2 + bx + c. In this chapter you will learn to form quadratic equations from word problems and solve them by factorisation, by splitting the middle term into two parts whose product is ac. You will then use the quadratic formula, x = (-b plus or minus sqrt(b^2 - 4ac))/(2a). The discriminant D = b^2 - 4ac decides the nature of the roots: two distinct real roots if D > 0, two equal real roots if D = 0 and no real roots if D < 0. You will use it to decide whether a situation, such as a park of given perimeter and area, is possible.

Detailed Worksheet: Quadratic Equations Section A - Definitions (10 marks) 1. Define a quadratic equation and write its standard form. Which coefficient must be non-zero, and why? (2 marks) 2. What is meant by a root of a quadratic equation? How are the roots of ax^2 + bx + c = 0 related to the zeroes of the polynomial ax^2 + bx + c? (2 marks) 3. State the quadratic formula for the roots of ax^2 + bx + c = 0 and the condition under which it gives real roots. (2 marks) 4. Define the discriminant of a quadratic equation. State the nature of the roots when D > 0, D = 0 and D < 0. (2 marks) 5. Explain the method of splitting the middle term with the help of the equation 2x^2 - 5x + 3 = 0. (2 marks) Section B - Calculations and Applications (15 marks) 6. Check whether (x - 2)(x + 1) = (x - 1)(x + 3) is a quadratic equation. Give reasons. (3 marks) 7. Find two consecutive positive integers whose product is 306. (3 marks) 8. Find the roots of 2x^2 - 7x + 3 = 0 using the quadratic formula and verify by factorisation. (3 marks) 9. Find the values of k for which the equation kx(x - 2) + 6 = 0 has two equal roots. Also find the roots. (3 marks) 10. Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2? If so, find its length and breadth. (3 marks) Section C - Diagrams (10 marks) 11. Draw a right triangle whose altitude is 7 cm less than its base and whose hypotenuse is 13 cm. Taking the base as x cm, label the sides, form the quadratic equation and find the other two sides. (4 marks) 12. Draw a rectangle representing a plot whose length is one more than twice its breadth and whose area is 528 m^2. Label the sides in terms of x, form the equation and find the dimensions. (3 marks) 13. Draw the graph of y = x^2 - 5x + 6 for x = 0 to 5 and read the roots of x^2 - 5x + 6 = 0 from the graph. Verify by factorisation. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Is the following situation possible? If so, determine their present ages: the sum of the ages of two friends is 20 years, and four years ago the product of their ages in years was 48. Explain your conclusion using the discriminant. (5 marks) 15. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Form the quadratic equation, find the speed of the train, and explain why one root is rejected. (5 marks) 16. Find the nature of the roots of the following equations and find the roots if they are real: (i) 2x^2 - 3x + 5 = 0 (ii) 3x^2 - 4sqrt(3)x + 4 = 0 (iii) 2x^2 - 6x + 3 = 0. What does each result mean graphically? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write each equation in standard form before solving. Reject roots that are not meaningful in the given situation and give a reason.