Математика20 мин

Nonlinear Equations

Equations with squares, square roots, or absolute values — substitution and squaring tactics.

Three core tactics

**Isolate the radical or square**, then square both sides — and check for extraneous roots.
**u-substitution** for hidden quadratics: $u = x^2$ turns $x^4 - 5x^2 + 4 = 0$ into $u^2 - 5u + 4 = 0$ .
**Casework on absolute values**: split into two equations.

Squaring can introduce false solutions. Always plug back into the original.

Square root

Solve $\sqrt{x + 7} = x - 1$ .

Square: $x + 7 = (x-1)^2 = x^2 - 2x + 1$ .
$0 = x^2 - 3x - 6$ . (or rearrange to $x^2 - 3x - 6 = 0$ )
Quadratic formula: $x = (3 \pm \sqrt{33})/2$ .
Reject negative root since $x - 1 \ge 0$ required: $x = (3 + \sqrt{33})/2 \approx 4.37$ .

Жауап: $x = \dfrac{3 + \sqrt{33}}{2}$