MATH 101EQUATIONS IN ONE VARIABLE

Lecture by: Ms. Cherry EstabilloALGEB-X: REAL NUMBER SYSTEM

ALGEB-X: REAL NUMBER SYSTEMEQUATION

- a mathematical statement that denotes equality between two expressionsAny number that satisfies the equation is called a root or solution of the equation.

ALGEB-X: REAL NUMBER SYSTEMTYPES OF EQUATIONConditional Equation the statement of equality is true only for SOME of real number Example:

ALGEB-X: REAL NUMBER SYSTEMTYPES OF EQUATIONIdentity Equation true for ALL real numbers Example:

ALGEB-X: REAL NUMBER SYSTEMTYPES OF EQUATIONContradiction the statement is NOT satisfied for any real number (i.e. no solution)Example:

ALGEB-X: REAL NUMBER SYSTEMLinear Equation in One Variable- Any equation that can be written in the form

where a and b are real constants and x is a variableExamples:

ALGEB-X: REAL NUMBER SYSTEMAddition Property of EquationsUse these properties to solve linear equations.Example: Solve x 5 = 12.x 5 = 12 x 5 + 5 = 12 + 5 x = 17 Original equation The solution is preserved when 5 is added to both sides of the equation.

That is, the same number can be added to or subtracted from each side of an equation without changing the solution of the equation.If a = b, then a + c = b + c and a c = b c.

ALGEB-X: REAL NUMBER SYSTEMMultiplication Property of EquationsIf a = b and c 0, then ac = bc and . That is, an equation can be multiplied or divided by the same nonzero real number without changing the solution of the equation.Example: Solve 2x + 7 = 19.2x + 7 = 19 2x + 7 7 = 19 7 2x = 12 x = 6 Original equation The solution is preserved when 7 is subtracted from both sides. Simplify both sides. 6 is the solution.

ALGEB-X: REAL NUMBER SYSTEMExample 1Solve:SOLUTION

ALGEB-X: REAL NUMBER SYSTEMExample 2Solve:SOLUTION

ALGEB-X: REAL NUMBER SYSTEMFRACTIONAL EQUATIONS- A fractional equation is an equation containing fractions.How do we solve this?Eliminate the denominator by multiplying both sides of the equation by the LCD.

Always check for extraneous roots.

ALGEB-X: REAL NUMBER SYSTEMFRACTIONAL EQUATIONSExamples.1.)

2.)

3.)

LITERAL EQUATIONS- A literal equation is an equation that involves more than one variable.How do we solve this?Our goal is to solve the equation for a specific variable. The rest of the letters are to be taken as constants.

ALGEB-X: REAL NUMBER SYSTEMLITERAL EQUATIONSExamples.1.)

2.)

3.)

ALGEB-X: REAL NUMBER SYSTEMQUADRATIC EQUATION in one variable x is an equation that can be expressed in the form

where a,b and c are real numbers with a0.The imaginary number:

QUADRATIC EQUATIONS

ALGEB-X: REAL NUMBER SYSTEMCASE 1: b=0Finding the root by Extracting the Square RootSQUARE ROOT PROPERTY:

If ,then where c is a constant.Examples

ALGEB-X: REAL NUMBER SYSTEMCASE 2: c=0-Use FactoringExamples

OR

ALGEB-X: REAL NUMBER SYSTEMCASE 3: 1.) Factoring2.) Completing the Squares3.) Quadratic Formula

ALGEB-X: REAL NUMBER SYSTEMFinding the root by FactoringZERO FACTOR PROPERTY:

If a, b are real numbers with ab=0, then a=0 or b=0.

ALGEB-X: REAL NUMBER SYSTEMFinding the root by FactoringEXAMPLES1.)

2.)

ALGEB-X: REAL NUMBER SYSTEMExample 1Find the solution set:SOLUTIONor or SS: {-5/2,7/3}

ALGEB-X: REAL NUMBER SYSTEMExample 2Find the solution set:SOLUTIONor or SS: {4,5}

ALGEB-X: REAL NUMBER SYSTEMFinding the root by Completing the SquareTo complete the square of a quadratic form , add the square of one-half the coefficient of x; that is, add

Thus, becomes

ALGEB-X: REAL NUMBER SYSTEMFinding the root by Completing the SquareEXAMPLES1.)

2.)

ALGEB-X: REAL NUMBER SYSTEMExample 1:SOLUTIONRemember that

ALGEB-X: REAL NUMBER SYSTEMExample 1:SOLUTIONor

ALGEB-X: REAL NUMBER SYSTEMFinding the root by the Quadratic Formula

If , a0, then

ALGEB-X: REAL NUMBER SYSTEMExample 1:SOLUTIONa=2, b=5, c=3or

ALGEB-X: REAL NUMBER SYSTEMExample 2:SOLUTIONa=3, b=-2, c=-10

ALGEB-X: REAL NUMBER SYSTEMDISCRIMINANT AND NATURE OF ROOTSDiscriminant :D = b2 4ac, if ax2 + bx + c = 0

DiscriminantNature of Rootsb2 4ac > 0if a perfect squareif not a perfect square2 Rational (REAL) and unequal roots2 Irrational (REAL) and unequal rootsb2 4ac = 01 Rational (REAL) and equal rootsb2 4ac < 02 Imaginary and unequal roots

ALGEB-X: REAL NUMBER SYSTEMFind the discriminant and discuss the nature of roots.EXAMPLE 1SOLUTIONa=1, b=-1, c=2There are two imaginary roots for this equation.

ALGEB-X: REAL NUMBER SYSTEMFind the discriminant and discuss the nature of roots.EXAMPLE 2SOLUTIONa=1, b=-6, c=8There are two real (RATIONAL) roots for this equation.

ALGEB-X: REAL NUMBER SYSTEMFind the discriminant and discuss the nature of roots.EXAMPLE 3SOLUTIONa=3, b=-2, c=-2There are two real (IRRATIONAL) roots for this equation.

ALGEB-X: REAL NUMBER SYSTEMFRACTIONAL EQUATIONS LEADING TO QUADRATIC EQUATIONSExamples1.)

2.)

3.)

ALGEB-X: REAL NUMBER SYSTEMExample 1:Find the solution setSOLUTIONThe LCD is: xoror

ALGEB-X: REAL NUMBER SYSTEMCheck:Correct!SS:{-2,-3}Correct!

ALGEB-X: REAL NUMBER SYSTEMExample 2:Find the solution setSOLUTIONThe LCD is: x - 1or

ALGEB-X: REAL NUMBER SYSTEMCheck:Correct!SS:{-1}NO! ER.

ALGEB-X: REAL NUMBER SYSTEMRADICAL EQUATIONSSome examples are :

To solve a radical equation, we may use

an equation in which a variable occurs in a radical 1.) If a=b, then

2.)

ALGEB-X: REAL NUMBER SYSTEMRADICAL EQUATIONSEXAMPLES1.)

2.)

3.)

4.)

ALGEB-X: REAL NUMBER SYSTEMExample 1:Find the solution setSOLUTION( )2Check:Correct!SS:{33}

ALGEB-X: REAL NUMBER SYSTEMExample 2:Find the solution setSOLUTION( )2or

ALGEB-X: REAL NUMBER SYSTEMCheck:Correct!SS:{5}NO! ER.

ALGEB-X: REAL NUMBER SYSTEMExample 3:Find the solution setSOLUTION

ALGEB-X: REAL NUMBER SYSTEMExample 3:Find the solution setSOLUTIONorCheck:NO! ER.NO! ER.SS:{ }

ALGEB-X: REAL NUMBER SYSTEMEQUATIONS in QUADRATIC FORMExamples1.)

2.)

3.)

ALGEB-X: REAL NUMBER SYSTEM