Weekly Assignment in Quadratic Equations

Flashcards based on mathematical problems and conditions for quadratic equations found in the weekly assignment transcript.

Value of $\frac{1}{a} + \frac{1}{b}$ for roots $a$ and $b$ of the equation $x^2 - px + q = 0$

$$\frac{p}{q}$$

Value of $\frac{1}{\alpha^3} + \frac{1}{\beta^3}$ for roots $\alpha, \beta$ of the equation $x^2 + 2x + 4 = 0$

$$\frac{1}{4}$$

Condition where the sum of roots of $ax^2 + bx + c = 0$ equals the sum of their squares

$$b(a + b) = 2ac$$

Value of $\lambda$ when the sum of roots equals their product for $\lambda x^2 + 2x + 3\lambda = 0$

None of these (Calculated as $\lambda = -\frac{2}{3}$)

Value of $(\alpha + 1) (\beta + 1)$ for roots $\alpha$ and $\beta$ of the equation $x^2 - a(x + 1) - b = 0$

$$1 - b$$

Value of $m$ if the product of roots of $mx^2 + 6x + (2m - 1) = 0$ is $-1$

$$\frac{1}{3}$$

Condition where the difference between corresponding roots of $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is the same and $a \neq b$

$$a + b + 4 = 0$$

Quadratic equation with roots $\frac{1}{3 + \sqrt{2}}$ and $\frac{1}{3 - \sqrt{2}}$

$$7x^2 - 6x + 1 = 0$$

Values of $p$ and $q$ if they are the roots of $x^2 + px + q = 0$

$$p = 1, q = -2$$

Value of $b^2 - 4c$ when factors of $x^2 - bx + c = 0$ are two consecutive integers

$$1$$

Product of the roots when roots are equal in magnitude but opposite in sign for $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$

$$- \left( \frac{p^2 + q^2}{2} \right)$$

Value of $\frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b}$ for roots $\alpha, \beta$ of $ax^2 + bx + c = 0$

$$-\frac{2}{a}$$

Condition for reciprocal roots in the equation $\frac{x - m}{mx + 1} = \frac{x + n}{nx + 1}$

$$n = 0$$