1Cademy - Square Root Property

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Quadratic Equation

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Square Root Property

The Square Root Property provides a direct method for solving quadratic equations of the form $x^2 = k$ . If $x^2 = k$ and $k \geq 0$ , then:

$x = \sqrt{k} \quad \text{or} \quad x = -\sqrt{k}$

This can be written more compactly as $x = \pm\sqrt{k}$ . The property follows from the definition of a square root: since squaring and taking a square root are inverse operations, if $n^2 = m$ then $n$ is a square root of $m$ . Because both a positive number and its negative counterpart have the same square, the property always produces two solutions — the principal square root of $k$ and its opposite.

When $k$ is a perfect square (such as $9$ , $16$ , or $25$ ), the equation $x^2 = k$ can also be solved by rewriting in standard form and factoring as a difference of squares. However, when $k$ is not a perfect square (such as $7$ ), factoring is not possible, and the Square Root Property becomes essential. In that case, the solutions are left in radical form — for example, $x^2 = 7$ gives $x = \pm\sqrt{7}$ .

The condition $k \geq 0$ is essential. When $k < 0$ , the equation $x^2 = k$ has no real solution, because the square root of a negative number is not a real number. For instance, if isolating the quadratic term yields $q^2 = -24$ , then applying the property gives $q = \pm\sqrt{-24}$ , but $\sqrt{-24}$ is not real, so no real value of $q$ satisfies the equation.

The property extends naturally to equations in which the squared expression is an entire binomial rather than a single variable. An equation of the form $(x - h)^2 = k$ is solved by treating the binomial $(x - h)$ as the quadratic term: applying the property gives $x - h = \pm\sqrt{k}$ , and adding $h$ to both sides yields $x = h \pm \sqrt{k}$ . The same conditions apply — $k$ must be non-negative for real solutions to exist. When $k$ is a perfect square, the solutions are rational numbers; when $k$ is not a perfect square, the solutions contain simplified radicals.

Updated 2026-04-21

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