Equations in Linear and Quadratic Forms At the end of this section, you should be able to solve equations that are convertible to equations in linear or quadratic forms: Equations involving rational expressions Equations involving radicals Equations in quadratic forms
Equations Involving Rational Expressions We consider equations involving fractions where the variable appears in the numerator or denominator or both. We clear our equation with fractions by multiplying both sides of the equation with the LCD of the fractions and solving the resulting linear or quadratic equation. We must take note that the LCD should not be equal to zero.
Equations Involving Rational Expressions Steps in solving an equation involving fractions: 1) Find the LCD of the fractions and multiply both sides of the equation by this LCD. ) Solve for the resulting linear or quadratic equation. 3) Check if the solutions found in step satisfies the original equation. 4) Only include those solutions that satisfy the equation in the Solution Set.
Equations Involving Rational Example 1:Solve for x in Solution: The LCD of the RE is Multiplying both sides by the LCD: Expressions 1 7 x x x x 1 x x1. 1 7 1 1 x x 1 x x x x x x x1 x 7 x x
Equations Involving Rational Expressions Checking the results shows that the LCD 0 for x Therefore, the solution set is SS
Equations Involving Rational Expressions Ex : Solve for x in Solution: 3 3 x x x x 3 5 6 x3 x The LCD of the fractions is. Multiplying both sides by the LCD yields: 3 x x 3 3 3x 6 x 6 3 x 3.
Equations Involving Rational Expressions Checking the results shows that the LCD = 0 for x 3. Thus, 3 is NOT a solution, hence, there is NO SOLUTION to the equation. Therefore, the solution set is SS
Equations Involving Radicals RULE: If a = b, then a = b. Conversely, if a = b, then a = b or a = b. Thus, squaring both sides of the equation may result to an answer that is actually NOT a root (solution) of the original equation. This is called an extraneous root. CHECKING IS NECESSARY!!!
Equations Involving Radicals Steps in solving an equation involving radicals: 1) Isolate the radical in one side of the equation. ) Raise both sides of the equation by an exponent that will remove the radical (i.e., if the index of the radical is, then square both sides of the equation). 3) Repeat steps 1 and to remove all the radicals. 4) Solve the resulting linear or quadratic equation. 5) Check for extraneous roots. These solutions will not be included in the solution set.
Equations Involving Radicals Example 1: Solve for x in x 1 Solution: Squaring both sides of the equation yields: x 14 x 5 Check: Substitute the value of x to the original equation. 5 1 4.
Equations Involving Radicals Thus, x = 5 is the solution and the solution set is SS 5.
Equations Involving Radicals Example : Solve for x in x 3 x 3 Solution: First isolate the radical by transposing x to the right side and then squaring both sides of the equation yields: x 3 3 x x 3 9 6x x x 8x 1 0 x 6x 0 x 6 or x
Equations Involving Radicals Check: Substitute the values of x to the original equation. Case 1: Case : If x 6, then 6 3 6 1 3 6 9 6 3 6 9 3 If x, then 3 4 3 1 1 3 Thus, the solution is x = and SS.
Equations Involving Radicals Try these on the board: (Volunteers please ) 1. x-1 x. 5x 3 x 7 3. x x 5
Equations in Quadratic Form These are equations that are of the form a b c 0 The quantity inside the parentheses may be any variable or algebraic expression. We treat ( ) as a variable and solve this quadratic equation, and then solve for the variable in ( ).
Equations in Quadratic Form Steps in solving an equation in quadratic form: 1) Write the given equation in quadratic form (i.e., a( ) + b( ) + c = 0). ) Solve for ( ) using the methods in solving quadratic equations (by factoring, completing the square or quadratic formula). 3) Solve for the value of the variable in ( ).
Equations in Quadratic Form Example 1: Solve for x in x 5x 4 0. 4 Solution: The equation can be written as: x 5 x 4 0 Quadratic in x x x 4 1 0 x or x 1 x x SS,,1, 1. 4 or 1
Equations in Quadratic Form Example : Solve for x in 1 3x 4x 1 0. Solution: The equation can be written as: 1 1 x x 1 1 x x 3 4 1 0 1 or 1 3 1 1 x x 3 1 1 0 1 1 1 or 1 x 3 x x 3 or x1 SS 3,1.
Equations in Quadratic Form Try these on the board: (Volunteers please ) 1.) x /3 3x 1/ 3 9 0.) 4 x - 1 x 3 x - 1 x 10
Additional Examples Find the solution set. 1) ) 3) 1 3 6x x x x 4 1 3 4 x x x x 6 3x 15 3 6 x x 3 x 3x
Additional Examples Find the solution set. 1) x1 3x1 ) x 6 x 3) x x 5 1 3 4) x 8 3 0
Additional Examples Find the solution set. 1 3 3 1) 6x 7x 3 0 4 ) x 18x 0 4 3) 3 x st 10 x st 8 0, solve for t
For Fun Find the value of x. 1.) x 1 1 1 1 1 1....) x...
Section Summary You solved equations that are convertible to equations in linear or quadratic forms: Equations involving fractions Equations involving radicals Equations in quadratic forms
Algebraic equations Notice that not all algebraic equations are solvable in R but are solvable in C. C is algebraically closed.