Quadratic Equations and Expressions

Notes on Quadratic Equations and Expressions

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  • Created on: 25-09-12 17:49
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IGCSEAdditional MathematicsQuadratic Equations and Expressions
Maximum/ Minimum value of a Quadratic Equation
A quadratic expression such as 2x2- 8x+ 11 or -4x2 +12x -8 has either a maximum or a minimum value.
Let's find out how we can find the extreme values for any quadratic expressions.
The expression ax2+bx+c can also be written as y =a (x-h)2+ k . This form of expression is called
complete square form.
Now how do we know whether the expression has a maximum value or a minimum value?
Writing a quadratic expression in complete square form helps us find out.
y = a(x - h)2 + k
a < 0the equation has a maximum value
a > 0the equation ha a minimum value
(h, k)turning point (it will be explained later)
kthis is the maximum or minimum value
Example: y = 2(x - 3)2 + 1
a = 2, h = 3, k = 1
a > 0 therefore, the expression has a minimum value
Since k = 1, the expression has a minimum value of 1.
You also need to know about what type of curve you will get when it is a minimum value and what type
of curve you will get when it is a maximum value.
A minimum value curve will look something like this:
1 | PageIGCSE2012

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
A maximum value curve will look something like this:
Roots of a Quadratic Equation
Sometimes, you might not get a whole number when solving an equation using "middle- term break". To
get an accurate value, there is a very useful special formula called the "quadratic formula".
ax2 + bx + c = 0 can be written as
x = -b±2
2
b -4ac
a
This is the quadratic formula.
Let's see how this is possible.…read more

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
2 2
(x + (2b b -4ac
a )) = 4a2
±(b2-4ac)
x + 2b
a = 2a
±(b2-4ac)
x =- 2b
a 2a
x = -b±2
2
b -4ac
a (P roved )
Solving Quadratic Inequality:
There are three possible cases for a quadratic inequality root.
b2- 4ac > 0 the quadratic inequality has real and distinct roots.
b2- 4ac = 0 the quadratic inequality has two real and equal roots.…read more

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
Its maximum value curve will look something like this.
a< 0
b2- 4ac = 0 The quadratic inequality has no real roots (imaginary roots).
It has only one x-intercept and the x-axis is the tangent to the parabola.…read more

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
Its maximum value curve will look something like this.
a<0
=
b2- 4ac < 0 The quadratic inequality has no real roots (imaginary roots).…read more

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
Its maximum value curve will look something like this.
a<0
How to solve a quadratic inequality:
Example: (x - 1)(x - 4) < 0
Step 1: Factorize and write down the x-intercept values like this.
x - 1 < 0 or x - 4 < 0
x < 1 or x < 4
Step 2: Find whether it has maximum value or a minimum value.…read more

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
2 2
[(x2 - 25 5 5
2x + ( 2) ) - ( 2) ] + 4 = 0
2
1(x - 5 25
2) - 4 + 4 = 0
You do not need to do the rest. We just need to know the max/min value.
So, we now know that the curve has a minimum value (a=1)
Step 3: Plot it.
A minimum value curve forms a parabola. Now you can plot it. We know the x-intercepts.…read more

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IGCSEAdditional MathematicsQuadratic Equations and Expressions
Since 4 is not less than 0 x - 1 > 0
Step 5: Write down the range:
1<x<4 This is your solution
8 | PageIGCSE2012…read more

Comments

jack unsworth

really good!

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