214 algebra De mystif ieD
Quadratic equations, to be studied in Chapter 10, have their variables
squared—that is, the only powers on variables are one and two. Some quadratic
equations are equivalent to linear equations. Once each side is simplified, the
terms containing x^2 “cancel,” that is, they add to 0.EXAMPLE
Solve the equation.(6x – 5)^2 = (4x + 3)(9x – 2)
(6x − 5)(6x − 5) = (4x + 3)(9x − 2) Begin with the FOIL method.
36 x^2 − 30x − 30x + 25 = 36x^2 − 8x + 27x − 6
36 x^2 − 60x + 25 = 36x^2 +19x − 6Because 36x^2 is on each side of the equation, they cancel each other, and
we are left with − 60x + 25 = 19x − 6, an ordinary linear equation.−+=−
++
=−
++
==60 25 19 6
60 60
25 79 6
66
31 79
31
79xx
xx
xxxBecause we neither multiplied (nor divided) both sides by an expression
involving a variable nor raised both sides to a power, it is not necessary to
check the solution. For the sake of accuracy, however, checking solutions
is a good habit.PRACTICE
Solve the equation.1.^185
32
7
3−
+x=
x2.^6
52
9
xx− 76=
+EXAMPLE
Solve the equation.EXAMPLE
Solve the equation.PRACTICE
Solve the equation.PRACTICE
Solve the equation.