PleasantCarnelian264
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## 5 Questions

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 $P$?

$P \leq 4$

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

• 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.

• 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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