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Questions and Answers
Which one of these is the value of $a$ in the equation $x^3 + ax^2 + b = 0$ if it has a double root?
Which one of these is the value of $a$ in the equation $x^3 + ax^2 + b = 0$ if it has a double root?
- $27a + 4b^3 = 0$
- $4a + 27b^3 = 0$ (correct)
- $4a^3 + 27b = 0$
- None of these
If the equation $(x - a)(x - 10) + 1 = 0$ has integral roots, what are the possible values of $a$?
If the equation $(x - a)(x - 10) + 1 = 0$ has integral roots, what are the possible values of $a$?
- 8, 10 (correct)
- 10, 12
- None of these
- 12, 8
What is the other root of the equation $ix^2 - 2(i + 1)x + (2 - i) = 0$ if one root is $2 - i$?
What is the other root of the equation $ix^2 - 2(i + 1)x + (2 - i) = 0$ if one root is $2 - i$?
- $i$
- $2 - i$
- $-i$ (correct)
- $2 + i$
How many solutions does the equation $(\sin(ex))^2 = 5x + 5^{-x}$ have?
How many solutions does the equation $(\sin(ex))^2 = 5x + 5^{-x}$ have?
For which values of $k$ does the equation $3x^2 + 2x(k + 1) + (k^2 - 3k + 2) = 0$ have roots of opposite signs?
For which values of $k$ does the equation $3x^2 + 2x(k + 1) + (k^2 - 3k + 2) = 0$ have roots of opposite signs?
Flashcards
Double Root
Double Root
A polynomial equation has a double root if a root appears twice. In this case, the discriminant of the quadratic must be equal to zero. The discriminant of $ax^2 + bx + c = 0$ is $b^2 - 4ac$.
Quadratic Formula
Quadratic Formula
The roots of a quadratic equation can be determined using the quadratic formula. The quadratic formula states that the roots of $ax^2 + bx + c = 0$ are $x = (-b b^2 - 4ac) / 2a$.
Sum and Product of Roots
Sum and Product of Roots
For a quadratic equation with roots $r_1$ and $r_2$, the sum of the roots is $-b/a$ and the product of the roots is $c/a$.
Integral Roots
Integral Roots
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Complex Number
Complex Number
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