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a.math quadratic
發問:
A=alpha B=beta 1.if the equations x^2+2x-2k and x^2-8x+(8k-20)=0 have a common root A , find the value of k and A 2.it is given that that A and B are the roots of the equation x^2+px+q=0, and A-3B and B-3A are the roots of the equation x^2-12x+q=0, where p and q are constants. find p and q
最佳解答:
1.α2 + 2α- 2k = 0 --- ( 1 ) α2 - 8α + (8k-20) = 0 --- ( 2 ) ( 1 ) – ( 2 ) : 10α – 10k + 20 = 0 α = k – 2 --- ( 3 ) Put ( 3 ) into ( 1 ). (k–2)2 + 2 (k–2)–2k = 0 k2 – 4k + 4 + 2k – 4 – 2k = 0 k ( k – 4 ) = 0 k = 0 or 4 α = 0 – 2 = - 2 or 4 – 2 = 2 2.Since a、b are the roots of the equation x2 + px + q = 0 a + b = -p/1 a + b = -p ..... (1) ab = q/1 ab = q ..... (2) Since a–3b、b–3a are the roots of the equation x2–12x + q = 0 (a–3b) + (b–3a) = -(-12)/1 a–3b + b–3a = 12 -2a–2b = 12 a + b = -6 ..... (3) 由 (1) 及 (3)， -p = -6 p = 6 ..... (4) Since a–3b、b–3a are the roots of the equation x2 - 12x + q = 0 (a–3b)(b–3a) = q/1 ab–3a2 – 3b2 + 9ab = q -3(a2 + b2) + 10ab = q -3(a2 + 2ab + b2) + 6ab + 10ab = q -3(a+b)2 + 16ab = q -3(-6)2 + 16(q) = q 【By (2), (4)】 -3(36) + 16q = q -108 = -15q q = 36/5
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