Equations Quadratic in Form Review - Intermediate Algebra...

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Revised04/101 HFCC Math Lab Intermediate Algebra 21 EQUATIONS QUADRATIC IN FORMDefinition:An equation which is not quadratic but can be transformed into a quadratic equation by using a suitable substitution, is called an equation quadratic in form. Such equations can be easily recognized by noticing that the power of an expression in one term is twice the power of the same expression in another term. Examples: 1. 63980xxis an equation quadratic in form because the power 6is twice the power3. 2. 21335140yyis an equation quadratic in form because the power 23is twice the power 13. 3. 42680xxis an equation quadratic in form because the power 4 is twice the power 2. The standard form of an equation quadratic in form is given by 20rraxbxc, where ris any rational number and 0a. 2Procedure for solving 0rraxbxcStep 1. Let rxuso that 222rrxxu. Step 2. Make these substitutions in the given equation. The given equation becomes 20aubucwhich is a quadratic equation in u. Step 3. Solve the quadratic equation of Step 2 by factoring (if possible) or by using the quadratic formula 242bbacua. Step 4. Replace uby rxin the solution(s) obtained for uin Step 3. Step 5. Solve the equation(s) of Step 4 for xby the methods already known to you.
Revised04/102 Please study the following examples very carefully and make sure you understand each and every step. Ex. 1: Solve 42680xxRewrite the given equation as 222680Equation 1xxLet 2.xuEquation (1) becomes 2680uuFactor the left side. 240uuHence, 20uor 40u2uor 4uReplace uby 2.x22xor 24xExtracting the roots, we get 2xor 4x2xor 2xThus the four solutions are 2,2,2, 2xEx. 2: Solve 2133253xxRewrite the given equation as 21133253Equation 1xxLet 13.xuEquation (1) becomes 2253uuSubtract 3 from both sides 22530uuFactor the left side. 3210uuHence, 30uor 210u3uor 21u
Revised04/103 33333313or 2127or8xxxx3uor 12uReplace uby 13x. 133xor 1312xRemember that 133.xxSo, we have 33xor 312xCube both sides of both equations. Thus the two solutions are 127,8xxEx. 3: Solve 46xxSince 144xxand x=12x, we can write 24xxThe given equation can be rewritten as 2446Equation 1xxLet 4.xuEquation (1) becomes 26uuSubtract 6 from both sides.

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