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Paper 1 Algebra. Leaving Certificate Helpdesk 20 th 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: - PowerPoint PPT Presentation
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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 1Solve 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)2 𝑥2 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 1The 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: