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Questions and Answers
What is the solution of the equation $2x^2 - 9 = 0$?
What is the solution of the equation $2x^2 - 9 = 0$?
- 3, -3 (correct)
- 3
- -3
- 0
If $P$ is a constant and the equation $Px^2 + 4x + 1 = 0$ has real roots, what is the possible range of values for $P$?
If $P$ is a constant and the equation $Px^2 + 4x + 1 = 0$ has real roots, what is the possible range of values for $P$?
- $P < 4$
- $P \leq 4$ (correct)
- $P = 4$
- $P > 4$
If $p$ and $q$ are the roots of the equation $3x^2 + 7x + 3 = 0$, what is the value of $p \cdot q$?
If $p$ and $q$ are the roots of the equation $3x^2 + 7x + 3 = 0$, what is the value of $p \cdot q$?
- 9 (correct)
- $-\frac{7}{3}$
- $-1$
- None of the above
If one root of the equation $ax^2 + 10x + 5 = 0$ is three times the other, what is the value of $a$?
If one root of the equation $ax^2 + 10x + 5 = 0$ is three times the other, what is the value of $a$?
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If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 3ax + 2a^2 = 0$ and $\alpha^2 + \beta^2 = 5$, what is the value of $a$?
If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 3ax + 2a^2 = 0$ and $\alpha^2 + \beta^2 = 5$, what is the value of $a$?
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Study Notes
Quadratic Equations
- The equation $2x^2 - 9 = 0$ has a solution.
- The equation $Px^2 + 4x + 1 = 0$ has real roots if $P$ is a constant.
- The possible range of values for $P$ is determined by the existence of real roots.
Roots of Quadratic Equations
- The roots of the equation $3x^2 + 7x + 3 = 0$ are $p$ and $q$.
- The value of $p \cdot q$ is determined by the equation.
Relationships Between Roots
- If one root of the equation $ax^2 + 10x + 5 = 0$ is three times the other, then the value of $a$ is determined.
- The equation $x^2 + 3ax + 2a^2 = 0$ has roots $\alpha$ and $\beta$.
- The value of $a$ is determined by the condition $\alpha^2 + \beta^2 = 5$.
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