Substitution To Solve Equations
Gbogbo ọrọ náà
Some equations look frightening until you notice a hidden quadratic inside them. By naming a repeated chunk with a new letter, you can turn a quartic, or an equation in powers of \(e^x\), into a plain quadratic you already know how to solve. This change of variable is one of the most elegant tricks in algebra.
In this lesson you will learn to spot a substitution such as \(u=x^2\), solve the resulting quadratic, and then return to the original variable to finish. The classic example is \(x^4-5x^2+4=0\), which becomes \(u^2-5u+4=0\) and unlocks four solutions.
Ebumnobi
- Use a substitution to form and solve a quadratic equation in order to solve a related equation.
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Ekele diri gi maka imecha ihe karịrị na Substitution To Solve Equations. Ugbu a na ị na-enyochakwa isi echiche na echiche ndị dị mkpa, ọ bụ oge iji nwalee ihe ị ma. Ngwa a na-enye ụdị ajụjụ ọmụmụ dị iche iche emebere iji kwado nghọta gị wee nyere gị aka ịmata otú ị ghọtara ihe ndị a kụziri.
Ị ga-ahụ ngwakọta nke ụdị ajụjụ dị iche iche, gụnyere ajụjụ chọrọ ịhọrọ otu n’ime ọtụtụ azịza, ajụjụ chọrọ mkpirisi azịza, na ajụjụ ede ede. A na-arụpụta ajụjụ ọ bụla nke ọma iji nwalee akụkụ dị iche iche nke ihe ọmụma gị na nkà nke ịtụgharị uche.
Jiri akụkụ a nke nyocha ka ohere iji kụziere ihe ị matara banyere isiokwu ahụ ma chọpụta ebe ọ bụla ị nwere ike ịchọ ọmụmụ ihe ọzọ. Ekwela ka nsogbu ọ bụla ị na-eche ihu mee ka ị daa mba; kama, lee ha anya dị ka ohere maka ịzụlite onwe gị na imeziwanye.
- Using u = x^2, the equation x^4 - 5x^2 + 4 = 0 becomes: A. u^2 - 5u + 4 = 0 B. u^2 + 5u + 4 = 0 C. u - 5u + 4 = 0 D. 2u - 5 = 0 Answer: A
- How many real solutions has x^4 - 5x^2 + 4 = 0? A. 1 B. 2 C. 3 D. 4 Answer: D
- Solve x^4 - 5x^2 + 4 = 0. A. x = 1 or x = 4 B. x = +/-1 or x = +/-2 C. x = +/-1 or x = +/-4 D. x = 2 only Answer: B
- For e^{2x} - 3e^x + 2 = 0, the substitution u = e^x gives the values: A. u = 1 or u = 2 B. u = -1 or u = -2 C. u = 0 or u = 3 D. u = 2 only Answer: A
- Solve e^{2x} - 3e^x + 2 = 0. A. x = 0 or x = ln 2 B. x = 1 or x = 2 C. x = ln 2 only D. x = -1 or x = -2 Answer: A
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