A cubic curve passes through the points (2,3) and (−3,−22). Find the equation of the curve if it has a stationary point at (1,−6).

1. Set up the general equation for a cubic curve and find its derivative.

Let the equation of the cubic curve be $y = ax^3 + bx^2 + cx + d$.

Differentiating with respect to $x$:
$\frac{dy}{dx} = 3ax^2 + 2bx + c$

2. Use the stationary point $(1, -6)$ to form two equations.

Since the point $(1, -6)$ lies on the curve, we substitute $x=1, y=-6$ into the curve equation:
$a(1)^3 + b(1)^2 + c(1) + d = -6$
$a + b + c + d = -6 \quad \text{--- (1)}$

Since $(1, -6)$ is a stationary point, $\frac{dy}{dx} = 0$ at $x = 1$:
$3a(1)^2 + 2b(1) + c = 0$
$3a + 2b + c = 0 \quad \text{--- (2)}$

3. Use the points $(2, 3)$ and $(-3, -22)$ to form two more equations.

Substituting $(2, 3)$ into the curve's equation:
$a(2)^3 + b(2)^2 + c(2) + d = 3$
$8a + 4b + 2c + d = 3 \quad \text{--- (3)}$

Substituting $(-3, -22)$ into the curve's equation:
$a(-3)^3 + b(-3)^2 + c(-3) + d = -22$
$-27a + 9b - 3c + d = -22 \quad \text{--- (4)}$

4. Eliminate $d$ to form a system of equations in terms of $a$, $b$, and $c$.

From (1), make $d$ the subject:
$d = -6 - a - b - c$

Substitute $d$ into (3):
$8a + 4b + 2c + (-6 - a - b - c) = 3$
$7a + 3b + c = 9 \quad \text{--- (5)}$

Substitute $d$ into (4):
$-27a + 9b - 3c + (-6 - a - b - c) = -22$
$-28a + 8b - 4c = -16$
Divide by $-4$:
$7a - 2b + c = 4 \quad \text{--- (6)}$

5. Solve the reduced system for the remaining constants.

Subtract (6) from (5) to eliminate $a$ and $c$:
$(7a + 3b + c) - (7a - 2b + c) = 9 - 4$
$5b = 5 \implies b = 1$

Substitute $b = 1$ into (2) and (5):
From (2): $3a + 2(1) + c = 0 \implies c = -3a - 2 \quad \text{--- (7)}$
From (5): $7a + 3(1) + c = 9 \implies c = 6 - 7a \quad \text{--- (8)}$

Equating (7) and (8):
$-3a - 2 = 6 - 7a$
$4a = 8 \implies a = 2$

Substitute $a = 2$ back into (7):
$c = -3(2) - 2 = -8$

Substitute $a, b,$ and $c$ into the expression for $d$:
$d = -6 - 2 - 1 - (-8) = -1$

6. Write the final equation of the cubic curve.

With $a = 2$, $b = 1$, $c = -8$, and $d = -1$, the equation of the cubic curve is:
$y = 2x^3 + x^2 - 8x - 1$