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第 9 週 · 二次方程 · 第 4 節

根與系數的關係 · 由根造方程

不用真把根解出來，光看係數就知道兩根「加起來多少、乘起來多少」。這節把和積公式、對稱式、造方程從零講透 — DSE 乙部高頻。

▶ 第一關： 直接報出「和」與「積」
a x² + b x + c = 0abcsum α + β = −b ÷ aprod αβ = c ÷ akeep the minus signe.g. x² − 4x − 3 = 0 : sum = −(−4) = 4, prod = −3read the coefficients — no need to solve for the roots
sealed parcelα , β inside(do NOT open)→read labeltotal weight α+β = 4total volume αβ = −3x² − 4x − 3 = 0 → sum / product read straight off, parcel never opened
▶ 第二關： 對稱式化簡（不解根）
symmetric expr → write with S = α+β and P = αβα² + β²= S² − 2P(α − β)²= S² − 4P1/α + 1/β= S / Pα²β + αβ²= P · Swatch: α²+β² uses −2P, (α−β)² uses −4P (different!)translate to S, P first — then plug in the numbers from Part 1
worked plug-in: x² − 4x − 3 = 0 (S = 4, P = −3)S = 4P = −3α² + β²= S² − 2P= 16 − 2(−3)= 22(α − β)²= S² − 4P= 16 − 4(−3)= 281/α + 1/β= S / P= 4 ÷ (−3)= −4/3minus times minus → plus : −2(−3) = +6, −4(−3) = +12
▶ 第三關： 由根造方程
build an equation from its rootsknow two roots→sum = α+β product = αβ(transform first if asked)↓x² − ( sum ) x + ( product ) = 0mnemonic: x² minus-sum plus-product
⚔ BOSS： 三招綜合 + DSE 最值題
DSE 2017 P1 Q18 type : parabola meets line, then minimise (a−b)²y = 19ab|a − b|(a − b)²= (a+b)² − 4ab= k² + 4k + 23= (k+2)² + 19(k+2)² ≥ 0 ⇒min (a−b)² = 19at k = −2