Slide 1:ALGEBRA LECTURE NOTES & PROBLEM SETS: PART 6


Slide 2:QUADRATIC EQUATION IT IS AN EQUATION OF THE SECOND DEGREE IN THE FORM OF: WHERE a, b, AND c ARE CONSTANTS WITH a  0. IT IS A COMPLETE QUADRATIC IF b  0. IT IS A PURE QUADRATIC IF b = 0. TO SOLVE THE ROOTS OF A QUADRATIC EQUATION: FACTORING COMPLETING THE SQUARE QUADRATIC FORMULA


Slide 3:SOLUTION OF A QUADRATIC EQUATION BY FACTORING TRANSPOSE TERMS TO OBTAIN ZERO AS ONE MEMBER; OBTAINING THE EQUATION IN THE FORM OF f(x) = 0. FACTOR THE f(x) IF POSSIBLE. PLACE EACH FACTOR TO BE EQUAL TO ZERO. SOLVE FOR x.


Slide 4:SOLUTION OF A QUADRATIC EQUATION BY COMPLETING THE SQUARE TRANSPOSE ALL TERMS INVOLVING x TO THE LEFT-HAND SIDE AND ALL OTHER TERMS TO THE RIGHT-HAND SIDE. COMBINE SIMILAR TERMS. DIVIDE BOTH MEMBERS BY THE NUMERICAL COEFFICIENT OF x2. THIS IS NORMALIZING THE EQUATION. COMPLETE THE SQUARE ON THE LEFT-HAND SIDE BY ADDING THE SQUARE OF ONE-HALF OF THE ABSOLUTE VALUE OF THE NUMERICAL COEFFICIENT OF x TO BOTH SIDES OF THE EQUATION.


Slide 5:SOLUTION OF A QUADRATIC EQUATION BY COMPLETING THE SQUARE REWRITE THE LEFT-HAND SIDE AS THE SQUARE OF A BINOMIAL. EXTRACT THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION; USING A DOUBLE SIGN ON THE RIGHT-HAND SIDE.


Slide 6:COMPLEX NUMBERS: AN OVERVIEW A COMPLEX NUMBER IS THE SUM OF A REAL NUMBER AND AN IMAGINARY NUMBER. AN IMAGINARY NUMBER IS A MULTIPLE OF i OR j, WHERE i OR j IS THE SQUARE ROOT OF ‑1. COMPLEX NUMBERS CAN BE EXPRESSED IN THE FOLLOWING FORMS:  RECTANGULAR FORM: BY DEFINITION OR WHERE a AND b ARE REAL NUMBERS, EITHER POSITIVE OR NEGATIVE INTEGERS. POLAR FORM: WHERE r IS THE RADIUS, ALWAYS A POSITIVE INTEGER, AND Ө IS ANGLE (COMMONLY) IN DEGREES. EXPONENTIAL FORM: WHERE r IS THE RADIUS, ALWAYS A POSITIVE INTEGER, AND Ө IS ANGLE (COMMONLY) IN RADIANS.


Slide 7:COMPLEX NUMBERS: AN OVERVIEW


Slide 8:COMPLEX NUMBERS: AN OVERVIEW


Slide 9:SOLUTION OF A QUADRATIC EQUATION BY USING THE QUADRATIC FORMULA CLEAR THE EQUATION OF FRACTIONS, IF THEY ARE PRESENT. WRITE THE EQUATION IN THE STANDARD FORM. EXTRACT THE VALUES OF a, b, AND c. SUBSTITUTE THE VALUES OF a, b, AND c IN THE QUADRATIC FORMULA TO OBTAIN THE ROOTS.


Slide 10:SOLUTION OF A QUADRATIC EQUATION BY USING THE QUADRATIC FORMULA


Slide 11:EXPONENTIAL EQUATION IT IS AN EQUATION IN WHICH THE VARIABLE APPEARS AS AN EXPONENT OR A PART OF AN EXPONENTIAL EXPRESSION. THEOREM: IF bx = by AND b  0, THEN x = y, PROVIDED THAT b > 0.


Slide 12:RADICAL EQUATION IT IS AN EQUATION INVOLVING A VARIABLE WITH A FRACTIONAL EXPONENT OR AN EQUATION IN WHICH A VARIABLE APPEARS UNDER A RADICAL SIGN. RULE: IF TWO NUMBERS ARE EQUAL, THEN THEIR SQUARES ARE EQUAL. IF a = b, THEN a2 = b2. STEPS TO FOLLOW IN SOLVING A RADICAL EQUATION: ARRANGE THE TERMS SO THAT ONE TERM WITH A RADICAL IS BY ITSELF ON ONE SIDE OF THE EQUATION.


Slide 13:RADICAL EQUATION STEPS TO FOLLOW IN SOLVING A RADICAL EQUATION (CONTINUED): SQUARE BOTH SIDES OF THE EQUATION. COMBINE LIKE TERMS. IF A RADICAL REMAINS, REPEAT ALL THE PRECEDING STEPS. SOLVE FOR THE RESULTING ROOT OF THE VARIABLE. CHECK APPARENT SOLUTIONS IN THE ORIGINAL EQUATION.


Slide 14:RADICAL EQUATION