U38 More about Equations: Essential Formulas

Substitution Method

For equations of the form $a[f(x)]^2 + b[f(x)] + c = 0$, where $f(x)$ is an expression in $x$, we can use the substitution $y = f(x)$. This transforms the equation into a standard quadratic equation in $y$: $ay^2 + by + c = 0$. Solve for $y$, then back-substitute to find the solutions for $x$.

Substitution from Linear Equation

Given a system: one linear equation (e.g., $ax + by + c = 0$) and one non-linear equation (e.g., $px^2 + qxy + ry^2 + sx + ty + u = 0$). Solve the linear equation for one variable (e.g., $y = mx + k$) and substitute into the non-linear equation. This yields a single-variable equation (often quadratic) which can be solved.

Intersection of Graphs

Solving a pair of simultaneous equations graphically corresponds to finding the intersection points of their respective graphs. The number of real solutions equals the number of intersection points.

For a system involving a quadratic curve and a straight line, the discriminant $\Delta$ of the resulting quadratic equation after substitution determines the number of intersections:

• $\Delta > 0$: Two distinct intersections (two real solutions).
• $\Delta = 0$: One intersection (tangent line, one repeated real solution).
• $\Delta < 0$: No intersection (no real solutions).