Quadratic Equations
This chapter covers solving quadratic equations by factorisation and using the quadratic formula, and determining the nature of roots using the discriminant.
An equation of the form ax^2 + bx + c = 0, where a, b, c are real numbers and a is not 0, is called a quadratic equation. It is called quadratic because the highest power of the variable x is 2.
The roots of ax^2 + bx + c = 0 are given by x = (-b +/- root(b^2 - 4ac)) / 2a. The term D = b^2 - 4ac is called the discriminant. If D > 0, there are two distinct real roots. If D = 0, there are two equal real roots. If D < 0, there are no real roots.
Q. Solve the quadratic equation x^2 - 7x + 12 = 0 by factorisation.
We need two numbers that multiply to 12 and add to -7: these are -3 and -4. So x^2 - 7x + 12 = (x - 3)(x - 4) = 0, giving x = 3 or x = 4.
Not every quadratic equation can be solved by simple factorisation, especially when roots are irrational; in such cases, the quadratic formula always works.
Q. Find the discriminant of the equation 2x^2 - 4x + 3 = 0 and determine the nature of its roots.
Here a = 2, b = -4, c = 3. Discriminant D = b^2 - 4ac = (-4)^2 - 4(2)(3) = 16 - 24 = -8. Since D < 0, the equation has no real roots.
If the discriminant of a quadratic equation is 0, the equation has:
Correct answer: (B) Two equal real roots
The standard form of a quadratic equation is ax^2 + bx + c = 0, where a is _______ (cannot equal).
Not equal to zero (a ≠ 0)
Every quadratic equation has at least one real root.
False — a quadratic equation has no real roots when its discriminant (b^2 - 4ac) is negative, since the square root of a negative number is not a real number.
The sum of the squares of two consecutive positive integers is 365. Find the integers by forming and solving a quadratic equation.
Let the integers be x and x+1. x^2 + (x+1)^2 = 365, giving 2x^2 + 2x + 1 = 365, so 2x^2 + 2x - 364 = 0, or x^2 + x - 182 = 0. Factoring: (x - 13)(x + 14) = 0, so x = 13 (rejecting the negative root). The integers are 13 and 14.