equations 5.3 notes - mastering-mathematics
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EQUATIONS
OUTLINE
ā¢ Linear Equations
ā¢ Literal Equations
ā¢ Quadratic Equations
ā¢ Cubic Equations
ā¢ Linear Inequalities
ā¢ Simultaneous Equations
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LINEAR EQUATIONS
ā¢ solve complex linear equations involving algebraic fractions
LINEAR EQUATIONS
Linear equations have exactly one solution.
eg. Solve 2š„ 1 3 š„ 1
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EQUATIONS WITH FRACTIONS
Solve:
š„ 13
š„2
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PRONUMERALS ON THE BOTTOM
When pronumerals are on the bottom, we treat the fractions exactly
the same.
2š„
32š„
11š„
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6
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BINOMIAL NUMERATORS OR DENOMINATORS
Solve:
š„ 23
š„ 54
BINOMIAL NUMERATORS OR DENOMINATORS
Solve
š„š„ 1
3š„ 1
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LITERAL EQUATIONS
ā¢ change the subject of formulas
LITERAL EQUATIONS
A literal equation has more than one variable. We cannot solve a literal
equation unless we do simultaneous equations. We can change the
subject of the equation.
Make y the subject of 2š„ 3š¦ 5.
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LITERAL EQUATIONS
We can change the subject of a formula to make it easier to solve
questions.
Find the value of h if š 120 and š 6 by changing the subject of the formula š šš ā.
LITERAL EQUATIONS
Make r the subject of š“ šš .
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LITERAL EQUATIONS
Make š the subject of šš
1 š.
QUADRATIC EQUATIONS
ā¢ solve equations of the form šš„ šš„ š 0 by factorisation and by 'completing the squareā
ā¢ use the quadratic formula to solve quadratic equations
ā¢ identify whether a given quadratic equation has real solutions, and whether or not they are equal
ā¢ solve a variety of quadratic equations and check the answers through substitution
ā¢ substitute a pronumeral to simplify higherāorder equations in order to solve them
ā¢ solve quadratic equations resulting from substitution into formulas or through solving problems and check their solutions
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QUADRATIC EQUATIONS
Recall that quadratic equations may have one, two or no solutions.
š„ 0
š„ 1
š„ 9
FACTORISING AND SOLVING
For quadratic trinomials, we can factorise and use the null factor law to
solve the equation.
š„ 3š„ 2 0Null Factor Law:
If š“ šµ 0 then
either š“ 0 or šµ 0.
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FACTORISING AND SOLVING
Solve the following equations:
š„ 2š„ 63 0
3š„ 5š„ 12 0
CHECKING SOLUTIONS
Like all equations, we can check if our solutions are correct by
substituting back into the equation.
3š„ 5š„ 12 0
š„ 3,43
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COMPLETING THE SQUARE
What happens if we cannot factorise?
Solve š„ 6š„ 5 0
EXAMPLE
Solve š„ 4š„ 6 0 by completing the square.
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THE QUADRATIC FORMULA
Solve the equation šš„ šš„ š 0.
THE QUADRATIC FORMULA
Given any quadratic equation in the form šš„ šš„ š 0 we can
find the solutions of the equation by using the formula:
š„š š 4šš
2š
The solutions to a quadratic equation are called the roots or zeroes. They are the xāintercepts of the graph of the equation.
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THE QUADRATIC FORMULA
Use the quadratic formula to find the roots of the equation š„ 6š„ 5 0.
THE QUADRATIC FORMULA
Find the roots of the equation to 3š„ š„ 18 0 to 3 significant
figures.
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EQUATIONS REDUCIBLE TO QUADRATICS
Solve š„ 13š„ 36 0
EQUATIONS REDUCIBLE TO QUADRATICS
By making an appropriate substitution, solve the equation:
2 4 5 2 2 0
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THE DISCRIMINANT
There is a quick way to determine whether a quadratic equation has
zero, one or two solutions. All we need to do is calculate the expression
under the square root in the quadratic formula.
š„š š 4šš
2šThis is called the discriminant and has the symbol ā.
THE DISCRIMINANT
Letās look at some examples:
a š„ 5š„ 3 0
b) š„ 6š„ 9 0
c) 2š„ 4š„ 7 0
š„5 25 4 3
25 13
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š„6 36 4 9
26 0
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š„4 16 4 2 7
2 24 40
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THE DISCRIMINANT
A quadratic equation has:
ā¢ Two real roots if ā 0
ā¢ One real root if ā 0
ā¢ No real roots if ā 0
EXAMPLE
Use the discriminant to find the number of solutions to the following
quadratic equations:
a 2š„ š„ 9 0 ā 1 4 2 9 71 none
b) š„ 5š„ 11 0 ā 5 4 1 11 69 two
c) 4š„ 12š„ 9 0 ā 12 4 4 9 0 one
d) š„ 6š„ 12 0 ā 6 4 1 12 84 two
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THE DISCRIMINANT
There is one other piece of information we can get from the
discriminant. We can tell whether a quadratic can be factorised or not.
eg. Find the discriminant of the following:
2š„ š„ 6 ā 1 4 2 6 49
3š„ š„ 5 ā 1 4 3 5 61
2š„ š„ 3 ā 1 4 2 3 25
Two of these can be factorised. Which ones?
If the discriminant is a perfect square, the answer will be _________.
EQUATIONS FROM FORMULAE
The cosine rule is a formula that finds unknown sides or angles of a non
right angled triangle:
š š š 2ššššš š¶
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WORD PROBLEMS
A skydiver jumps from a plane. The height above the ground after tseconds is given by ā 1900 5š” .
a) What height was the plane when the skydiver jumped?
b) Approximately how many seconds will it take for the skydiver to
reach the ground?
WORD PROBLEMS
A skydiver jumps from a plane. The height above the ground after tseconds is given by ā 1900 5š” .
c) How many seconds does it take for the skydiver to fall 1000m?
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WORD PROBLEMS
The product of two consecutive, positive integers is 342. What are the
numbers?
CUBIC EQUATIONS
ā¢ solve simple cubic equations of the form šš„ š, leaving answers in exact form and as decimal approximations
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CUBE ROOT
Unlike square root, when we take the cube root there is only one
solution.
Why?
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CUBIC EQUATIONS
The highest power of a cubic equation is 3.
A cubic equation can have one, two or three solutions.
Solve the following cubic equations:
The opposite operation of 3 is ā.
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LINEAR INEQUALITIES
ā¢ solve linear inequalities, including through reversing the direction of the inequality sign and graph the solutions
LINEAR INEQUALTIES
Recall the two rules for solving inequalities:
ā¢ if we turn the inequality around (that is, swap sides) the inequality sign flips
ā¢ if we multiply or divide by a negative the sign flips
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EXAMPLES
Solve, and graph the solution on a number line:
2š„ 1 š„ 2 2š„ 1 3š„ 1
HARDER INEQUALITIES
Inequalities can have three parts to them. For example:
2 š„ 1 5
This means that š„ 1 lies between 2 and 5.
We can solve this inequality by applying opposite operations to all
three sides.
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HARDER INEQUALITIES
Solve 1 1 2š„ 5.
SIMULTANEOUS EQUATIONS
ā¢ use analytical methods to solve simultaneous equations, where one
equation is nonālinear
ā¢ use graphical methods to solve simultaneous equations, where one
equation is nonālinear
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SIMULTANEOUS EQUATIONS
Recall that there are three methods of solving simultaneous equations:
ā¢ Elimination
ā¢ Substitution
ā¢ Graphical
When one of the equations is nonālinear, it is not practical to use the
elimination method.
SUBSTITUTION METHOD
Solve the following simultaneous equations:
š¦ š„ 3š„ 1
š¦ 2š„ 5
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SUBTITUTION METHOD
Solve the following simultaneous equations.
š¦2š„
š¦ š„ 1
GRAPHICAL METHOD
Solve the simultaneous equations by
graphing and finding the points of
intersection.
š¦2š„
š¦ š„ 1
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
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