UNIT 6 : Quadratic equations and inequalities
Key unit competence
Model and solve algebraically or graphically daily life problems using quadratic equations or inequalities.
Learning objectives
6.1 Introduction
In Senior 3, we learnt about quadratic equations and ways to solve them.
Activity 6.1
Carry out research to obtain the definition of quadratic equation. Discuss your findings with the rest of the class.
The term quadratic comes from the word quad meaning square, because the variable gets squared (like x2). It is also called an “equation of degree 2” because of the “2” on the x.
The standard form of a quadratic equation looks like this: ax2 + bx + c = 0
where a, b and c are known values and a cannot be 0.
“x” is the variable or the unknown.
Here are some more examples of quadratic equations:
2x2 + 5x + 3 = 0 In this one, a = 2, b = 5 and c = 3
x2 − 3x = 0 For this, a = 1, b = –3 and c = 0, so 1 is not shown.
5x − 3 = 0 This one is not a quadratic equation. It is missing a value in x2 i.e a = 0, which means it cannot be quadratic.
6.2 Equations in one unknown
A quadratic equation in the unknown x is an equation of the form ax2 + bx + c = 0, where a, b and c are given real numbers, with a ≠ 0. This may be solved by completing the square or by using the formula
• If b2 – 4ac > 0, there are two distinct real roots
• If b2 – 4ac = 0, there is a single real root (which may be convenient to treat as two equal or coincident roots)
• If b2 – 4ac < 0, the equation has no real roots.
We know that the quadratic equation is of the form:
Sum and product of roots
Activity 6.2
In pairs, form the quadratic equations that have 7 and –3 as roots.
Solving quadratic equations by factorizing
6.3 Inequalities in one unknown
The product ab of two factors is positive if and only if
(i) a > 0 and b > 0 or
(ii) a < 0 and b < 0.
Sign diagrams
Although this method is sound it is not of much practical use in more complicated problems. A better method which is useful in more complicated problems is the following which uses the ‘’sign diagram’’ of the product (x – 1)(x + 2).
(x – 1) (x + 2) > 0
The critical values are x = 1 and x = –2. (i.e., the values of x at which the factor is zero.)
The sign diagram of (x – 1) (x + 2) is thus:
Inequalities depending on the quotient of two linear factors
Solving general inequalities
The techniques illustrated in the previous pages can be used to solve complicated inequalities. There are also other techniques which may be used in special cases.
6.4 Parametric equations
In case certain coefficients of equations contain one or several letter variables, the equation is called parametric and the letters are called real parameters. In this case, we solve and discuss the equation (for parameters only).
Parametric equations in one unknown
If at least one of the coefficients a, b and c depend on the real parameter which is not determined, the root of the parametric quadratic equation depends on the values attributed to that parameter
6.5 Simultaneous equations in two unknowns
To solve simultaneous equations involving a quadratic equation we use substitution of one equation into the other.
6.6 Applications of quadratic equations and inequalities
Activity 6.3
In groups of five, research on the importance and necessity of quadratic equations and inequalities.
Quadratic equations lend themselves to modelling situations that happen in real life. These include:
• projectile motions
• the rise and fall of profits from selling goods
• the decrease and increase in the amount of time it takes to run a kilometre based on your age, and so on.
The wonderful part of having something that can be modelled by a quadratic is that you can easily solve the equation when set equal to zero and predict. If you throw a ball (or shoot an arrow, fire a missile or throw a stone) it will go up into the air, slowing down as it goes, then come down again. A quadratic equation can tell you where it will be at any given time.
