1. **State the problem:** We have two functions $f(x)=(x+5)^2$ and $g(x)=(x-7)^2$. Their tangents have gradients related by $m_f=1.5m_g$. We want to find the equations of the tangents that maximize the area of the triangle formed above the x-axis by these tangents. 2. **Find derivatives (gradients) of the functions:** $$f'(x)=2(x+5)$$ $$g'(x)=2(x-7)$$ These represent the slopes of the tangents at any point $x$. 3. **Set the gradient relation:** Given $m_f=1.5m_g$, substitute derivatives: $$2(x_f+5)=1.5 \times 2(x_g-7)$$ Simplify: $$x_f+5=1.5(x_g-7)$$ $$x_f+5=1.5x_g-10.5$$ $$x_f=1.5x_g-15.5$$ 4. **Write tangent line equations:** At points $x_f$ and $x_g$, tangent lines are: $$y_f=m_f(x - x_f) + f(x_f)$$ $$y_g=m_g(x - x_g) + g(x_g)$$ where $$m_f=2(x_f+5)$$ $$m_g=2(x_g-7)$$ 5. **Find x-intercepts of tangents:** Set $y=0$: For $f$ tangent: $$0 = m_f(x - x_f) + f(x_f)$$ $$m_f(x - x_f) = -f(x_f)$$ $$x = x_f - \frac{f(x_f)}{m_f}$$ Similarly for $g$ tangent: $$x = x_g - \frac{g(x_g)}{m_g}$$ 6. **Calculate the base of the triangle:** The triangle is formed by the two tangent lines and the x-axis. The base is the distance between the two x-intercepts: $$B = \left| \left(x_g - \frac{g(x_g)}{m_g}\right) - \left(x_f - \frac{f(x_f)}{m_f}\right) \right|$$ 7. **Calculate the height of the triangle:** The height is the vertical distance between the two tangent lines at the point where they intersect. Since the triangle is above the x-axis, the height is the difference in y-values at the intersection point of the tangents. 8. **Find intersection of tangents:** Set $y_f = y_g$: $$m_f(x - x_f) + f(x_f) = m_g(x - x_g) + g(x_g)$$ Solve for $x$: $$x = \frac{m_f x_f - m_g x_g + g(x_g) - f(x_f)}{m_f - m_g}$$ 9. **Calculate height $H$ at intersection:** Substitute $x$ into one tangent line: $$H = m_f(x - x_f) + f(x_f)$$ 10. **Area of triangle:** $$A = \frac{1}{2} B H$$ 11. **Express $A$ in terms of $x_g$ only:** Use $x_f=1.5x_g - 15.5$, and substitute all expressions for $f(x_f)$, $g(x_g)$, $m_f$, $m_g$, $B$, and $H$ in terms of $x_g$. 12. **Maximize $A$ by differentiating w.r.t $x_g$ and setting derivative to zero:** Solve $\frac{dA}{dx_g} = 0$ for $x_g$. 13. **Calculate corresponding $x_f$, $m_f$, $m_g$, and tangent equations:** Use found $x_g$ to get $x_f$, then gradients and tangent lines. **Final answer:** The tangent equations maximizing the triangle area are: $$y = 2(x_f+5)(x - x_f) + (x_f+5)^2$$ $$y = 2(x_g-7)(x - x_g) + (x_g-7)^2$$ where $x_f=1.5x_g - 15.5$ and $x_g$ is the solution to $\frac{dA}{dx_g}=0$ from step 12.