Quadratic Equations

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5 Questions

What is the solution of the equation $2x^2 - 9 = 0$?

3, -3

If $P$ is a constant and the equation $Px^2 + 4x + 1 = 0$ has real roots, what is the possible range of values for $P$?

$P \leq 4$

If $p$ and $q$ are the roots of the equation $3x^2 + 7x + 3 = 0$, what is the value of $p \cdot q$?

9

If one root of the equation $ax^2 + 10x + 5 = 0$ is three times the other, what is the value of $a$?

$\frac{4}{15}$

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 $a$?

$\pm \frac{1}{2}$

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 $\alpha^2 + \beta^2 = 5$.

Test your knowledge of quadratic equations with this quiz. From finding solutions to understanding the discriminant, this quiz covers various aspects of quadratic equations and their roots.

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