Solving by Square Roots

When a quadratic has no x term, do not factor. Just isolate the square and take roots.

x² = 49 → x = ±7. Two answers, because both 7 and −7 square to 49.

The ± is mandatory. Writing only x = 7 loses half the solution.

It works with a bracket too:
(x − 3)² = 25 → x − 3 = ±5 → x = 3 ± 5, giving x = 8 or x = −2.

If the isolated square equals a negative number, there is no real solution. x² = −9 has none, because no real number squares to a negative. Algebra 2 introduces imaginary numbers to handle that case.

For u² = k with k > 0 the solutions are u = ±√k. The ± is required because squaring is not injective on ℝ: both √k and −√k map to k. This is also why √(u²) = |u| rather than u.

The method applies directly whenever the quadratic can be written with a single squared expression and no linear term — that is, in vertex form a(x − h)² + k = 0. Solving gives x = h ± √(−k/a), showing that the roots are symmetric about the axis x = h.

For k = 0 there is one solution of multiplicity 2; for k < 0 there is no real solution, and the parabola does not meet the x-axis. Over ℂ the solutions are ±i√|k|.

Completing the square exists precisely to convert an arbitrary quadratic into this form, making this the underlying method rather than a special case.