More about Equations

One linear and one quadratic equation

1. Make $y$ (or $x$) the subject of the linear equation.
2. Substitute into the quadratic equation.
3. Solve the resulting quadratic, then find the other unknown from the linear equation.

$$\begin{cases} y = x + 1 \\ y = x^2 - 1 \end{cases} \Rightarrow x^2 - x - 2 = 0 \Rightarrow x = 2 \text{ or } x = -1$$

The solutions are $(2, 3)$ and $(-1, 0)$: the points where the line meets the curve.

Number of intersection points

After substituting, look at $\Delta$ of the resulting quadratic:

• $\Delta > 0$: the line cuts the curve at two points.
• $\Delta = 0$: the line is a tangent to the curve (one point).
• $\Delta < 0$: the line does not meet the curve.

Equations reducible to quadratic equations

Fractional equations: multiply through by the denominator, then check that no root makes a denominator zero.

$$x - 2 = \frac{3}{x} \Rightarrow x^2 - 2x - 3 = 0 \Rightarrow x = 3 \text{ or } x = -1$$

Exponential equations: spot a hidden quadratic with a substitution.

$$4^x - 6(2^x) + 8 = 0$$

Let $u = 2^x$, so $4^x = u^2$: $u^2 - 6u + 8 = 0$, giving $u = 2$ or $u = 4$, so $x = 1$ or $x = 2$.

Logarithmic equations: combine the logs, remove them, then reject any root that makes a log of a non-positive number.

Common mistake: forgetting to check roots. In log equations a root like $x = -5$ must be rejected because $\log(-5)$ is undefined.