2.3 一元二次方程式的根与系数的关系 The Relations of Roots and Coefficients of Quadratic Equations - 独中课程 | 线上补习

高中一 | 高级数学

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2.3 The Relations of Roots and Coefficients of Quadratic Equations 一元二次方程式的根与系数的关系

[1]

If $\alpha$ , $\beta$ are the roots of 3x $^2$ – 2x – 3 = 0, find the values of

若 $\alpha$ , $\beta$ 是 3x $^2$ – 2x – 3 = 0 的根, 求值

[a] $\frac{1}{\alpha}+\frac{1}{\beta}$

[b] $\alpha^2 + \beta^2$

[c] $\frac{\beta}{\alpha}+\frac{\alpha}{\beta}$

Answer:

(a) $\frac{2}{3}$

(b) $2\frac{4}{9}$

[2]

The equation 3x $^2$ + 10x + p = 0, has roots $\alpha$ and $\beta$ where $\alpha$ – $\beta$ = 2 $\frac{2}{3}$ , find the value of p.

方程式3x $^2$ + 10x + p = 0 有根 $\alpha$ 和 $\beta$ 其中 $\alpha$ – $\beta$ = 2 $\frac{2}{3}$ , 求 p的值.

Answer:

[3]

The equation 2x $^2$ – 2x + 3 = 0 has roots p and q and the equation x $^2$ – x + 2m = 0 has roots and . Find the values of k and m.

方程式2x $^2$ – 2x + 3 = 0 有根 p 和 q 及方程式 x $^2$ – x + 2m = 0 有根和 . 求 k 和 m的值.

Answer:

$1\frac{1}{2}$ , $\frac{3}{4}$

[4]

If the square of the difference of the roots of equation x $^2$ – mx +15 = 0 is 4, find the value of m.

若方程式x $^2$ – mx +15 = 0 的根的平方差是 4, 求 m 的值.

Answer:

$\pm 8$

[5]

If $\alpha$ and $\beta$ are the roots of equation 2x $^2$ – 3x+ 4 = 0, find the equation whose roots are

若 $\alpha$ 和 $\beta$ 是方程式2x $^2$ – 3x+ 4 = 0 的根, 求作方程式其根是

[a] $\alpha+\frac{1}{\alpha}$ , $\beta+\frac{1}{\beta}$

[b] $\alpha^2$ , $\beta^2$

Answer:

(a) 8x $^2$ – 18x + 13 = 0

(b) 4x $^2$ + 7x + 16 = 0

[6]

One root of the equation 2x $^2$ – x + c = 0 is twice the other. Find the value of c.

方程式2x $^2$ – x + c = 0 其中一根是另外一根的两倍. 求 c的值.

Answer:

$\frac{1}{9}$

[7]

One root of the equation 3x $^2$ – 3px + p $^2$ = p + 6 is twice of the other. Find the values of p.

方程式3x $^2$ – 3px + p $^2$ = p + 6 其中一根是另外一根的一半. 求 p的值.

Answer:

6, -3

[8]

Given that $\alpha$ and $\beta$ are the roots of the equation 2x $^2$ = 3x – 4,

已知 $\alpha$ 和 $\beta$ 是方程式 2x $^2$ = 3x – 4的根,

[a]

form an equation whose roots are $\alpha$ – $\beta$ and $\beta$ – $\alpha$ .

作方程式其根是 $\alpha$ – $\beta$ and $\beta$ – $\alpha$ .

[b]

show that 4 $\alpha^3$ = $\alpha$ – 12.

证明4 $\alpha^3$ = $\alpha$ – 12.

Answer:

(a) 4x $^2$ + 23 = 0