1Cademy - Solving $$x^2 - 48 = 0$$ Using the Square Root Property

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Square Root Property

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Solving $x^2 - 48 = 0$ Using the Square Root Property

Solve $x^2 - 48 = 0$ by following the four-step procedure for applying the Square Root Property.

Step 1 — Isolate the quadratic term. Add $48$ to both sides so the squared variable stands alone:

$x^2 = 48$

Step 2 — Apply the Square Root Property:

$x = \pm\sqrt{48}$

Step 3 — Simplify the radical. The largest perfect square factor of $48$ is $16$ . Rewrite $48$ as $16 \cdot 3$ and apply the Product Property:

$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$

Therefore $x = \pm 4\sqrt{3}$ .

Step 4 — Check both solutions by substituting into the original equation:

For $x = 4\sqrt{3}$ : $(4\sqrt{3})^2 - 48 = 16 \cdot 3 - 48 = 48 - 48 = 0$ ✓

For $x = -4\sqrt{3}$ : $(-4\sqrt{3})^2 - 48 = 16 \cdot 3 - 48 = 48 - 48 = 0$ ✓

The solutions are $x = 4\sqrt{3}$ and $x = -4\sqrt{3}$ . Unlike equations where the constant is a perfect square (such as $x^2 = 169$ ), this equation requires simplifying the radical because $48$ is not a perfect square. The initial rearrangement — adding $48$ to both sides — demonstrates why Step 1 (isolating the quadratic term) is needed before the Square Root Property can be applied.

Updated 2026-04-21

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OpenStax Elementary Algebra

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