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Title: Quadratic Equations
Description: Writing Quadratic Equations in one variable
Description: Writing Quadratic Equations in one variable
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Quadratic Equations in One Variable
Definition
A quadratic equation in x is any equation that may be written in the form
ax2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0
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Examples
x2 + 2x = 4 is a quadratic since it may be rewritten in the form ax2 + bx + c = 0 by
applying the Addition Property of Equality and subtracting 4 from both sides of =
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x2 - 3 = 0 is a quadratic since it has the form ax2 + bx + c = 0 with b=0 in this case
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The term 2/x is the
same as 2x-1, and quadratics do not have x raised to any power other than 1 or 2
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Solving Quadratic Equations – Method 1 - Factoring
The easiest way to solve a quadratic equation is to solve by factoring, if possible
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Write your equation in the form ax2 + bx + c = 0 by applying the Distributive
Property, Combine Like Terms, and apply the Addition Property of Equality to
move terms to one side of =
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Factor your equation by using the Distributive Property and the appropriate
factoring technique
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3
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This is possible because of the Zero Product
Law
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Also the first two terms must multiply out to 3x2
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(3x + 7)(x - 1) = 0 gives us middle products 7x and –3x adding up to 4x
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com – Permission Granted For Use and Modification For Non-Profit Purposes
By the Zero Product Law, we can state
3x + 7 = 0 and x-1 = 0
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3x + 7 = 0 → 3x = -7 → x = -7/ 3
x-1=0 → x=1
Solving Quadratic Equations – Method 2 – Extracting Square Roots
Extracting square roots is a very easy way to solve quadratics, provided the equation is
in the correct form
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Algebraically, we are taking square roots of both sides of the
equation as shown below and inserting the ± to account for both a positive and negative
case
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Example: Solve x2 = 9 by extracting square roots
Example: Solve (2x – 5)2 + 5 = 3
(2x – 5)2 + 5 = 3
Given
(2x – 5)2 = -2
Addition Property of Equality used to add –5 to both sides
2
√ (2x – 5) = ±√(-2)
Extract Square Roots
2x – 5 = ± i√2
Simplify Radicals and Apply Definition of “i”
2x = 5 ± i√2
Addition Property of Equality
x = (5 ± i√2) / 2
Division Property of Equality
Solving Quadratic Equations – Method 3 – Completing The Square
This method of solving quadratic equations is straightforward, but requires a specific
sequence of steps
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Isolate the x2 and x-terms on one side of = by applying the Addition
Property of Equality
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Apply the Division Property of Equality to divide all terms on both sides by
the coefficient on x2
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From MathMotivation
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Take ½ of the coefficient on x
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Add this square to
both sides using the Addition Property of Equality
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Square 4/6 to get (4/6) •(4/6) = 16/36 = 4/9
when reduced
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Factor the left side
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Solve by extracting square roots
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Also, we need this method to justify and derive the Quadratic Formula
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From MathMotivation
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This is left as an
exercise
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com – Permission Granted For Use and Modification For Non-Profit Purposes
Title: Quadratic Equations
Description: Writing Quadratic Equations in one variable
Description: Writing Quadratic Equations in one variable