방정식, 공식 및 항등식
피어슨 에덱셀 · International GCSE · Mathematics A 주제 2 5:36 영어 내레이션 · 영어 + 중국어 자막 burned-in
장
전사본
The Higher formulae sheet gives only the quadratic formula, so the rest you carry.
高级卷公式表只给出求根公式，其余都要自己记。
Multiply powers by adding indices, and a power of a power multiplies them.
同底数幂相乘，指数相加；幂的乘方，指数相乘。
A negative index is a reciprocal and a fractional index is a root.
负指数是倒数，分数指数是开方。
And proportion means y equals k times a power of x.
正比例就是 y 等于 k 乘以 x 的某次幂。
Every term is a power of 3, so only the indices move.
每一项都是 3 的幂，所以只有指数在变。
Multiplying adds them and dividing subtracts: n minus 2, plus n plus 5, minus 3n plus 10.
相乘时指数相加，相除时相减：(n − 2) + (n + 5) − (3n + 10)。
The right side is 1, which is 3 to the power 0.
右边是 1，即 3⁰。
So minus n minus 7 is 0, and n is minus 7.
所以 −n − 7 = 0，n = −7。
Draw the product as a rectangle x plus 8 by x minus 5.
把乘积画成一个 (x + 8) × (x − 5) 的长方形。
Its four parts are x times x, minus 5x, plus 8x and minus 40.
四块分别是 x·x、−5x、+8x 和 −40。
The two x terms collect to 3x, so the expansion is x squared plus 3x minus 40.
两个 x 项合并成 3x，所以展开式是 x² + 3x − 40。
For three brackets, expand two first, then multiply through.
三个括号时，先展开两个，再乘第三个。
Building V from r takes two steps: cube, then multiply by 4 pi over 3.
由 r 得到 V 有两步：先立方，再乘以 4π/3。
Undo them in reverse: multiply by 3, divide by 4 pi, then take the cube root.
按相反顺序撤销：乘以 3，除以 4π，再开立方。
The same idea squares first in T equals 2 pi root l over g.
同样的思路用在 T = 2π√(l/g) 上要先平方。
Check any rearrangement by substituting numbers into both versions.
代入数字到前后两个式子中检验变形。
y is proportional to x cubed, and y is 24 when x is 2.
y 与 x³ 成正比，x = 2 时 y = 24。
Write y equals k x cubed: 24 equals 8k, so k is 3.
写成 y = kx³：24 = 8k，所以 k = 3。
The curve y is 3x cubed passes through both points, and y is 81 when x is 3.
曲线 y = 3x³ 经过这两个点；x = 3 时 y = 81。
Inverse proportion uses k over x to a power instead.
反比例则用 k 除以 x 的幂。
Scale one equation to match a coefficient, then add or subtract.
先把一个方程乘一个数，使某个系数相同，再相加或相减。
Take 2 x plus 3 y equals 17 and 3 x minus 5 y equals 35.
取 2x 加 3y 等于 17 和 3x 减 5y 等于 35。
Multiply the first by 3 and the second by 2, so both start with 6 x.
第一个乘 3，第二个乘 2，两个都以 6x 开头。
The first becomes 6 x plus 9 y equals 51, the second 6 x minus 10 y equals 70.
第一个变成 6x 加 9y 等于 51，第二个变成 6x 减 10y 等于 70。
Subtracting, 19 y equals minus 19, so y is minus 1, and then x is 10.
相减得 19y 等于负 19，所以 y 等于负 1，进而 x 等于 10。
Each equation is a straight line, and the solution 10, minus 1 is the point where the two lines cross.
每个方程都是 一条直线，解 (10, 负 1) 就是两条直线的交点。
Write 4 minus 3x minus 5 divided by the fraction as one fraction.
把 4 − (3x − 5) ÷ 分式写成一个分式。
Factorise the quadratic first: 3x minus 5 times x plus 2.
先把二次式因式分解：(3x − 5)(x + 2)。
Dividing by a fraction multiplies by its reciprocal, and the whole factor 3x minus 5 cancels.
除以分式就是乘以它的倒数，整个因式 3x − 5 约去。
Then use a common denominator: 9 over x plus 2.
再通分：9/(x + 2)。
To prove 0.306 recurring is 34 over 111, name it x.
要证明 0.306 循环等于 34/111，先设它为 x。
The block 306 has three digits, so multiply by 1000: both numbers now share the same endless tail.
循环节 306 有三位，所以乘以 1000：两个数有同样的无限尾巴。
Subtract and the tail vanishes: 999x is 306, which simplifies to 34 over 111.
相减后尾巴消失：999x = 306，约分为 34/111。
Vertex form is a translation of a x squared, so the minimum can be read straight off.
顶点式是 a 乘 x 的平方经过平移得到的，所以最小值可以直接读出。
Halve the x coefficient inside the bracket.
在括号内取 x 系数的一半。
2 x squared minus 11 x plus 9 becomes 2 times x minus eleven quarters, squared, minus forty-nine eighths.
2x 的平方减 11x 加 9 变成 2 乘以 x 减四分 之十一的平方，再减八分之四十九。
The minimum point is at 2.75 and minus 6.125.
最小值点在 2.75 和负 6.125。
Under y equals f of x minus 3, the graph moves 3 to the right, so the minimum moves to 5.75 and minus 6.125.
在 y 等于 f(x 减 3) 下，图像向右平移 3，所以最小值点移到 5.75 和负 6.125。
Choose the route by whether it factorises, and always check by substitution.
根据能否因式分解来选择方法，并且总要代入检验。
x squared plus x minus 30 factorises to x plus 6, times x minus 5, so the roots are minus 6 and 5: the two places the graph crosses the x axis.
x 的平方加 x 减 30 分解为 (x 加 6)(x 减 5)，所以根是负 6 和 5：就是图像与 x 轴相交 的两个位置。
When factorising fails, use the formula: 2 x squared plus 6 x minus 1 equals zero gives x equals minus 3 plus or minus root 11, all over 2.
不能因式分解时用求根公式：2x 的平方加 6x 减 1 等于零，得 x 等于负 3 加减根号 11，再除以 2。
From a context, a lawn of width w and length w plus 3 with area 70 gives w is 7, because a width of minus 10 is rejected.
在情境中，宽为 w、长为 w 加 3、面积为 70 的草坪，得 w 等于 7，因为宽为负 10 要舍去。
Find the critical values first; the parabola's sign then decides inside or outside.
先求临界值，再由抛物线的正负决定取两根之间还是两根之外。
A linear inequality behaves like an equation, except that multiplying by a negative flips the sign: 7 minus x at most 5 gives x at least 2.
一次不等式和方程一样解，只是乘以负数时不等号要反向：7 减 x 小于等于 5，得 x 大于等于 2。
For 6 x squared minus 7 x minus 20 greater than zero, factorise to 3 x plus 4, times 2 x minus 5.
对于 6x 的平方减 7x 减 20 大于零，分解为 (3x 加 4)(2x 减 5)。
The critical values are minus four thirds and five halves.
临界值是负三分之四和二分之五。
The parabola opens upwards, so it is positive outside the roots: x less than minus four thirds, or x greater than five halves.
抛物线开口向上，所以在两 根之外为正：x 小于负三分之四，或 x 大于二分之五。
Check yourself.
自我检测。
8 to the two thirds is 4.
8^{2/3} = 4。
x plus 8 times x minus 5 is x squared plus 3x minus 40.
(x + 8)(x − 5) = x² + 3x − 40。
From v equals u plus a t, t is v minus u over a.
由 v = u + at 得 t = (v − u)/a。
The constant k is 3.
常数 k = 3。
And 7 minus x at most 5 flips to x at least 2.
7 − x ≤ 5 变号后得 x ≥ 2。