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If a square garden has an area of $49\text{ ft}^2$, you instantly know each side is $7\text{ ft}$ long. But what if we ask for all algebraic solutions to $x^2 = 49$?

Solving an equation **by inspection** means looking directly at its structure and stating the solutions mentally without writing out lengthy algebraic steps.

Because squaring means multiplying a number by itself, finding $x$ in $x^2 = c$ simply asks: *which numbers multiply by themselves to give $c$?*

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  <title>Solving Quadratics by Inspection</title>
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    <button class="btn-val" data-n="3" data-c="9">x² = 9</button>
    <button class="btn-val active" data-n="4" data-c="16">x² = 16</button>
    <button class="btn-val" data-n="5" data-c="25">x² = 25</button>
    <button class="btn-val" data-n="6" data-c="36">x² = 36</button>
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Every positive number has both a positive square root (the **principal root**) and an opposite negative square root.

Formally, **The Square Root Property** states that if $x^2 = k$ and $k > 0$, the equation has exactly two real solutions: the principal square root $x = \sqrt{k}$ and the negative square root $x = -\sqrt{k}$, concisely written as $x = \pm \sqrt{k}$.

What happens when an equation isn't in isolated form right away?

Consider the equation $x^2 - 64 = 0$. By mentally adding $64$ to both sides, you see $x^2 = 64$.

Since $8^2 = 64$ and $(-8)^2 = 64$, you can immediately state the solutions: $x = \pm 8$.

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The most common mistake when solving by inspection is writing only the positive root ($x = 7$) and forgetting the negative root ($x = -7$).

While geometric lengths like garden sides can only be positive, purely algebraic variables can always take negative values.

We know how to solve $x^2 = 49$, but what happens if the squared term equals a negative number, like $x^2 = -49$?

Multiplying any real number by itself always produces a non-negative result, meaning $x^2 = -49$ has no **real solutions**.

In upcoming lessons, you will discover how mathematicians invented imaginary numbers to solve equations just like $x^2 = -49$ using this exact same inspection mindset.

math

Solving Quadratics by Inspection. Solve simple quadratic equations like x² = 49 by recognizing the square root immediately.

Solving Quadratics by Taking Square Roots

Solve a Quadratic Equation Using Square Roots (Skill Practice)

Reasoning About Equations and Taking Square Roots (Algebra 1 / HSA.REI.B.4.b)

When teaching solving by inspection, emphasize algebraic reasoning over mechanical procedures. Begin with verbal prompts ('What number squared is 49?') before formalizing into $x = \pm\sqrt{49}$. Explicitly contrast geometric side lengths (always positive) with algebraic solutions (both signs). Connect this foundational skill directly to upcoming methods such as completing the square and solving quadratics with complex roots ($x^2 = -k$).

Solving Quadratics by Inspection

What is Solving by Inspection?

Isolating the Squared Term

The Missing Sign Trap