To find the value of $p$ so that each pair of equations has a common root, we need to set up the conditions where the common root satisfies both equations in each pair.

Part (a)

Given equations:

1. $4x^2 + px - 12 = 0$
2. $4x^2 + 3px - 4 = 0$

Let $\alpha$ be the common root.

Substitute $\alpha$ into both equations: $4\alpha^2 + p\alpha - 12 = 0 \quad \text{(1)}$ $4\alpha^2 + 3p\alpha - 4 = 0 \quad \text{(2)}$

Subtract equation (1) from equation (2) to eliminate $\alpha^2$: $(4\alpha^2 + 3p\alpha - 4) - (4\alpha^2 + p\alpha - 12) = 0$ $(3p\alpha - p\alpha) - (4 - (-12)) = 0$ $2p\alpha + 8 = 0$

Solve for $\alpha$: $\alpha = -\frac{8}{2p} = -\frac{4}{p}$

Substitute $\alpha = -\frac{4}{p}$ into one of the original equations, let's use equation (1): $4\left(-\frac{4}{p}\right)^2 + p\left(-\frac{4}{p}\right) - 12 = 0$ $4\left(\frac{16}{p^2}\right) - 4 - 12 = 0$ $\frac{64}{p^2} - 16 = 0$

Solve for $p$: $\frac{64}{p^2} = 16$ $p^2 = \frac{64}{16} = 4$ $p = \pm 2$

Part (b)

Given equations:

1. $2x^2 + px - 1 = 0$
2. $3x^2 - 2x - 5 = 0$

Let $\beta$ be the common root.

Substitute $\beta$ into both equations: $2\beta^2 + p\beta - 1 = 0 \quad \text{(3)}$ $3\beta^2 - 2\beta - 5 = 0 \quad \text{(4)}$

Multiply equation (3) by 3 and equation (4) by 2 to eliminate $\beta^2$: $6\beta^2 + 3p\beta - 3 = 0 \quad \text{(5)}$ $6\beta^2 - 4\beta - 10 = 0 \quad \text{(6)}$

Subtract equation (6) from equation (5): $(6\beta^2 + 3p\beta - 3) - (6\beta^2 - 4\beta - 10) = 0$ $3p\beta + 4\beta + 7 = 0$

Factor out $\beta$ from the terms involving $\beta$: $\beta(3p + 4) + 7 = 0$

Since $\beta$ is the common root, it must satisfy the equation: $\beta(3p + 4) = -7$

Substitute $\beta$ back into one of the original equations. From equation (3), solve for $p$.

Given that the equations do not provide a simple expression for $\beta$, we need more information or different methods to solve for $p$.

Would you like to go deeper into solving this, or have any other questions?

5 Related Questions:

1. What are the conditions for two quadratic equations to have a common root?
2. How do you find the common root of two quadratic equations?
3. What methods can be used to solve quadratic equations simultaneously?
4. How does the discriminant affect the roots of quadratic equations?
5. Can two different quadratic equations have more than one common root?

Tip:

When solving for a common root in two quadratic equations, always try to eliminate the squared term first, which simplifies the algebraic manipulation.