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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 a0.

Note that if a=0, then the equation would simply be a linear equation, not quadratic.

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 =.

(2+x)(3-x)=0 is a quadratic since it may be rewritten in the form ax2+bx+c=0
by applying the Distributive Property to multiply out all terms and then combining
like terms.

x2-3=0 is a quadratic since it has the form ax2+bx+c=0 with b=0 in this case.

3x2-2x+4=0 is not a quadratic since it has the term 2x. The term 2x is the
same as 2x-1, and quadratics do not have x raised to any power other than 1 or 2.

Just remember: Quadratics always have an x2 term, possibly an x-term, and

possibly a constant term! If your equation has an x2 term or will have an x2 term
after multiplying out, it may be a quadratic, provided the other terms fit the form.

Solving Quadratic Equations-Method 1 -Factoring

The easiest way to solve a quadratic equation is to solve by factoring, if possible.

Here are the steps to solve a quadratic by factoring:

1. Write your equation in the form ax2+bx+c=0 by applying the Distributive

Properly, Combine Like Terms, and apply the Addition Property of Equality to

move terms to one side of =.

2. Factor your equation by using the Distributive Property and the appropriate

factoring technique. Note: Any type of factoring relies on the Distributive Property.

3. Let each factor =0 and solve. This is possible because of the Zero Product Law.

Example: Solve (3x+4)x=7

(3x+4)x=7 Given

3x2+4x=7 by the Distributive Property

3x2+4x-7=0 by the Addition Property of Equality

Now, factor 3x2+4x-7=0

This factors as (3x+?)(x-?)=0 or (3x-?)(x+?)=0 where the two unknown
numbers multiply to-7 when we use the Distributive Property to multiply out.

Also the first two terms must multiply out to 3x2. The middle products must add
up to 4x.

(3x+7)(x-1)=0 gives us middle products 7x and -3x adding up to 4x.

By the Zero Product Law, we can state
3x+7=0 and x-1=0.

Solve these two equations by using the Addition Property of Equality and the
Division Property of Equality.

3x+7=0

3x = -7

x = -73

Also

x-1 = 0

i.e x=1