Quadratic Equation Reciprocal Roots

For a general quadratic equation $Ax^2 + Bx + C = 0$, let the roots be $\alpha$ and $\beta$. The product of the roots is given by the formula $\alpha \beta = \frac{C}{A}$.

In this problem, the roots are stated to be reciprocals of each other. Let the roots be $\alpha$ and $\frac{1}{\alpha}$.

Applying the product of roots formula:

$ \alpha \times \frac{1}{\alpha} = \frac{C}{A} $

Simplifying the left side gives:

$ 1 = \frac{C}{A} $

This condition implies that $A = C$.

Finding 'a' in $3x^2 + 5x - a = 0$

Compare the given equation $3x^2 + 5x - a = 0$ with the standard form $Ax^2 + Bx + C = 0$.

• The coefficient $A$ is $3$.
• The coefficient $B$ is $5$.
• The constant term $C$ is $-a$.

Determining the Value of 'a'

Using the condition $A = C$ derived from the reciprocal roots property:

$ 3 = -a $

To find the value of $a$, multiply both sides by $-1$:

$ a = -3 $

Thus, the value of $a$ is $-3$.