Algebraic techniques
Apply index laws to simplify expressions with integer indices, expand binomial products, factorise quadratics, rationalise surds, and use the discriminant to determine the nature and number of roots of a quadratic.
Worked examples
Simplifying an expression using index laws
Straightforward
Problem
Simplify .
1
Apply the power of a power law to the numerator's first factor.
2
Apply the multiplication law to multiply the two powers in the numerator.
3
Apply the division law to divide by the denominator.
Answer
Factorising a quadratic fully
Moderate
Problem
Factorise fully.
1
Identify and take out the highest common factor (HCF) of all three terms.
2
Factorise the trinomial by finding two numbers that multiply to and add to .
3
Write the factorised form of the trinomial.
4
Include the common factor to give the complete factorisation.
Answer
Using the discriminant to determine the nature of roots
Challenging
Problem
For the quadratic , calculate the discriminant and determine how many -intercepts the parabola has.
1
Identify the values of , and from .
2
Substitute into the discriminant formula .
3
Interpret the sign of . Since , the quadratic has no real roots.
4
State the geometric interpretation: the number of -intercepts equals the number of real roots.
Answer
. Since , the parabola has no -intercepts.
Practise
Q1·Straightforward
Simplify using index laws. What is the value of the exponent in the simplified expression?
Explanation
Using the index law : . The exponent is .
Q2·Straightforward
Simplify using index laws. What is the value of the exponent in the simplified expression?
Explanation
Using the index law : . The exponent is .
Q3·Straightforward
Simplify using index laws. What is the value of the exponent in the simplified expression?
Explanation
Using the index law : . The exponent is .
Q4·Straightforward
Expand . What is the constant term of the result?
Explanation
. The constant term is .
Q5·Moderate
Factorise into the form where . What is the value of ?
Explanation
We need two numbers that multiply to and add to : these are and . So . Since , we have .
Q6·Moderate
The expression can be written in the form . What is the value of ?
Explanation
The highest common factor of and is . Factorising: . Writing as gives .
Q7·Moderate
Rationalise the denominator of and write the result in the form . What is the value of ?
Explanation
Multiply by : . So .
Q8·Moderate
Rationalise the denominator of and write the result in the form . What is the value of ?
Explanation
Multiply by : . So .
Q9·Challenging
Calculate the discriminant for the quadratic equation . What is the value of ?
Explanation
Here , , . So . Since , the quadratic has two distinct real roots and the parabola crosses the -axis at two points.
Q10·Challenging
Calculate the discriminant for the quadratic equation . What is the value of ?
Explanation
Here , , . So . Since , the quadratic has exactly one repeated real root and the parabola touches the -axis at one point.
Q11·Challenging
Calculate the discriminant for the quadratic equation . What is the value of ?
Explanation
Here , , . So . Since , the quadratic has no real roots and the parabola does not intersect the -axis.
Q12·Challenging
The discriminant of is . What does this tell you about the graph of ?
Explanation
Since , the equation has exactly one repeated root. This means the parabola touches the -axis at exactly one point (the vertex lies on the -axis). In this case, is the repeated root since .