Paper 1Algebra

Leaving Certificate Helpdesk 20th September 2012

General Content for Algebra

• Simultaneous Equations• Modulus Equations • Inequalities • The Nature of Roots of a Quadratic Equation• Complex Numbers

Simultaneous Equations: Example 1

Solve the simultaneous equations:

_______________________________________Step 1: Eliminate one of the variables.

Step 2: Solve for either or using the following equations:

Step 3: Solve for by subbing for in the equation:

Step 4: Solve for using one of the original equations.

We know and

Answers:

Simultaneous Equations in Three Variables

Method:

• Select one pair of equations and eliminate one of the variables.

• Select another pair and eliminate the same variable.• Solve these two new equations simultaneously.• Use answers to find third variable.

Simultaneous Equations: Example 2

Solve the simultaneous equations08

082

xyx

yx

Simultaneous Equations: Example 3

2012 Paper 1

Q1(a)

Method:Turn the rational inequality into a quadratic inequality by multiplying both sides by a positive expression.

Example:Solve the inequality

Note: multiplying both sides by a squared value ensures that the inequality sign is not affected.

Rational Inequalities

212

xx

(2 𝑥−1)2𝑥

2 𝑥−1<−2(2𝑥−1)2

Complete all multiplication and tidy up the expression

Solve the Quadratic to find the roots so that we can sketch the graph of the quadratic.

Roots:

When is ?

Answer:

Modulus Equations / Inequalities

RxwherexxforSolve ,312:

Solution: Square both sides

Complete all multiplication and tidy up the expression:

Solve the quadratic to find the roots and sketch the curve:

Roots:

Where is ?

Answer:

The inequality is true when

The Nature of Roots of a Quadratic

Example: 2009 Question 2 (b)(i)

The Nature of Roots of a Quadratic

Two real roots:

Equal roots:

Note: Roots are real if

The Nature of Roots of a Quadratic

Imaginary Roots:

Quadratic Roots Example 1

The equation has equal roots. Find the possible values of k.

0)1(2 kxkkx

Solution: Equal roots:

Quadratic Roots Example 2

Sample Paper 2012

Paper 1

Q3

(a) If is a root then

Conclusion: is a root of

If is a root then is a factor of

Solution: Divide into

We now know:

Solve to find final two roots

Use

Roots:

The real root must be as we are told at the start that

Thus

are the imaginary roots

Therefore

Answer: