LINEAR AND QUADRATIC EQUATIONS

1. BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA
EQUATIONS
1. EQUATION
An equation is a mathematical statement that has two expressions separated by an equal sign,
it is an equality that is true for some values of the letters or variables.
EXAMPLE: x + 1 = 2 x = 1
The members of an equation are each of the expressions that appear on both sides of
the equal sign.
The terms are the addends within the members.
The unknowns are the letters that appear in the equation.
The solutions or roots are the values that the letters must take to make the equality is true.
EXAMPLE: 2x − 3 = 3x + 2 x = 5
2 · (5) − 3 = 3 · (5) + 2
− 10 −3 = −15 + 2
−13 = −13
The degree of an equation is the largest of the degrees of the monomials that its
members form.
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2. 2. TYPES OF EQUATIONS ACCORDING TO THEIR DEGREE
Degree Name Example
One Linear equation. 5x + 3 = 2x +1
Two Quadratic equation. 5x + 3 = 2x2
+ x
Three Cubic equation. 5x3
+ 3 = 2x + x2
Four Quartic Equation. 5x3
+ 3 = 2x4
+1
3. SOLVING LINEAR EQUATIONS
In general, to solve a linear equation you must follow five steps: (Attention to the third and fith
steps)
1. Remove parentheses.
2. Remove denominators.
3. Group the terms of x in one member and the independent terms in the other. You can
shift any positive or negative term with opposite sign to the other side of an equation.
4. Reduce similar terms.
5. Solve the unknown. A factor on one side of an equation becomes a divisor on the
other side, a divisor on one side of an equation becomes a factor on the other side.
EXAMPLE 1: If the two members are divided by two, an equivalent equation is obtained:
EXAMPLE 2: Group the similar terms and independents, and then add:
EXAMPLE 3:
Remove parentheses:
Group the terms and add:
If both sides of the equation are divided by 3, an equivalent equation is obtained:
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3. EXAMPLE 4:
Remove denominators by finding the least common multiple.
The common denominator divides each of the denominators, multiplying the quotient obtained by the corresponding
numerator.
Remove parentheses, group and add the similar terms:
Divide both sides by −2:
EXAMPLE 5:
Remove parentheses and simplify the fractions:
Remove denominators, group and add the similar terms:
EXAMPLE 6:
Remove brackets:
Remove parentheses:
Remove denominators:
Remove parentheses:
Group terms:
Add:
Divide the two members by −9:
4. SOLVING QUADRATIC EQUATIONS
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4. A quadratic equation is any expression of the form: ax2
+ bx + c = 0, a ≠ 0. If any of its
coefficients: b or c, or both, are zero, then we have an incomplete quadratic equation.
a) SOLVING INCOMPLETE QUADRATIC EQUATIONS:
Type Method to solve
ax2
= 0
The solution is x = 0.
EXAMPLE 1:
EXAMPLE 2:
ax2
+ bx = 0
The following steps must be followed:
1. Remove the common factor, x:
2. The 1st factor equals 0.
A solution is always x = 0.
3. The other solution was obtained by solving the linear equation resulting from
equating the 2nd factor to zero.
EXAMPLE 1:
ax2
+ c = 0
We find it is similar to a linear equation, so we use the same method
EXAMPLE 1:
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5. EXAMPLE 2:
b) SOLVING QUADRATIC EQUATIONS:
To solve quadratic equations we need to replace the numerical value of a, b and c in the
following formula:
EXAMPLE 1: a=1, b=-5 and c= 6
EXAMPLE 2: a= 2, b=-7 and c=3
EXAMPLE 3: If a<0, multiply two members by (−1).
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6. EXAMPLE 2:
b) SOLVING QUADRATIC EQUATIONS:
To solve quadratic equations we need to replace the numerical value of a, b and c in the
following formula:
EXAMPLE 1: a=1, b=-5 and c= 6
EXAMPLE 2: a= 2, b=-7 and c=3
EXAMPLE 3: If a<0, multiply two members by (−1).
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