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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…

Page 2

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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…

Page 3

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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…

Page 4

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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.

Its minimum value curve would be something…

Page 5

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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 no x- intercept and y= ax2+bx+c is either always positive or always negative

Its minimum value curve would be…

Page 6

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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 <…

Page 7

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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…

Page 8

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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





8 | PageIGCSE2012 ©

Comments

jack unsworth

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really good!

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