Substitution To Solve Equations
Akopọ
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.
Awọn Afojusun
- Use a substitution to form and solve a quadratic equation in order to solve a related equation.
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Oriire fun ipari ẹkọ lori Substitution To Solve Equations. Ni bayi ti o ti ṣawari naa awọn imọran bọtini ati awọn imọran, o to akoko lati fi imọ rẹ si idanwo. Ẹka yii nfunni ni ọpọlọpọ awọn adaṣe awọn ibeere ti a ṣe lati fun oye rẹ lokun ati ṣe iranlọwọ fun ọ lati ṣe iwọn oye ohun elo naa.
Iwọ yoo pade adalu awọn iru ibeere, pẹlu awọn ibeere olumulo pupọ, awọn ibeere idahun kukuru, ati awọn ibeere iwe kikọ. Gbogbo ibeere kọọkan ni a ṣe pẹlu iṣaro lati ṣe ayẹwo awọn ẹya oriṣiriṣi ti imọ rẹ ati awọn ogbon ironu pataki.
Lo ise abala yii gege bi anfaani lati mu oye re lori koko-ọrọ naa lagbara ati lati ṣe idanimọ eyikeyi agbegbe ti o le nilo afikun ikẹkọ. Maṣe jẹ ki awọn italaya eyikeyi ti o ba pade da ọ lójú; dipo, wo wọn gẹgẹ bi awọn anfaani fun idagbasoke ati ilọsiwaju.
- 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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