useful to solve many purely algebraic problems. Let's demonstrate the technique with an techniques we discussed in class, your instinct might be to expand the left side and combine (x- 1) - 14x - 11) + 30 - 2 - 22x + 121 -14x+ 151 + 3 - x - 36x + 305 and wa s poke seems a face paint as by fin ding two numbers that multiply to 306 occurrence is squared , and the second occurrence is multiplied by - 14 . We say that (x - 11) substitution may help simplify in Peyman . When Substitute out a common subexpression. This is still a quadratic equation, but it's easier to solve Solve in terms of u. property we need two numbers that multiply to 32 and add to - 14 , so they must be 306 and adding compare this to the problems of finding two numbers man , to New split the equation using the zero product property and solve each equation. Solve in terms dina shove with the girlable x. To solve in terms of x, simply replace win thed for solutions substitution. Using substitution added a few more steps, but greatly simplified each step . A Higher-Order Equation I mentioned in class that the x - 29x3 + 100=0 Sample s in nice, and quadratic techniques. You may want to compare what we do with Substitute out a common subexpression. Make the su itution ar = x' to transform the degree- equation x -212 + 100-0 I'mableyimo a degree 2 equation in the variable u. (Hint: If s' in t, then what Equation of a- 2 Solve in terms of u. Solve the resulting equation in terms of u. Solve in terms of x. Plug back in for ' = some number or 's another number Recall me somedoctoring not by using radicals. A Rational Equation substitution can also be used to solve some rational equations, such as the one below. - 22x + 121 143- X- 11 begin by completely factoring the denominators. (x - 11) - 143 x- 10 1 Substitute out a common subexpression. Make the substitution = x- 15 to turn the rational equ equation in u. (Hint: If the fraction y - 17 is , then what is (x -1179 in terms of up Equation of a Solve in terms of u. Solve the resulting equation in terms of u. 3 Solve in terms of x. Phos x - It back in for u. I'm * some number or - 13 * another number Salve these equations for *. You may want to take a reciprocal of both sides of the w - b, thena - A Non-Monic Equation ample, x' - 17x + 72 is monic, but Zx' - Lx + 1 is not. Consider the non-monic equation 10x - 13x - 3 =0. We saw two approaches to solving equations like this: the ac-method was our preferred of those methods could work for this equation. In this problem, we'll learn a third method. 101043 - 12x - 39 -0 10 - 103 - 1312- 30-0 1 Substitute out a common subexpression. What about 105, " * * Box . (Hing : If Box is s, then what is - 138x in terms of u? 2 Solve in terms of u. Solve the resulting equation in terms of u. 3 Solve in terms of x Plug Box back in for a. 10 = some number or 10x = another num quations for z. Just divide by 102