L1 · Solving x² = a: solutions as square roots

Learning goal

“I can solve an equation of the form x² = a, give both solutions as exact square roots, and state whether the equation has two solutions, one solution, or no real solution - and prove each solution by substitution.”

Quadratic equation — an equation whose highest power of the variable is 2, like x² = 25. (A linear equation has highest power 1 and exactly one solution; a quadratic can have more.)

Square root — √a is the number that, multiplied by itself, gives a: √25 = 5 because 5 x 5 = 25. The √ symbol always names the non-negative root.

Plus-or-minus (±) — shorthand for "both the positive and the negative value": x = ±5 means x = 5 or x = -5.

No real solution — the honest answer when no real number can make the equation true, as with x² = -9: squaring any real number gives 0 or a positive number, never a negative.

Worked example

Solve x² = 25. Step 1: Read the question: "Which numbers, multiplied by themselves, give 25?" Step 2: 5 x 5 = 25, so x = 5 is a solution. But (-5) x (-5) = 25 as well - a negative times a negative is positive - so x = -5 is ALSO a solution. Step 3: Write the solution set with both: x = 5 or x = -5, shortened to x = ±5. On the number line: two points, at 5 and at -5, the same distance from 0. Step 4: Verify BOTH in the original: 5² = 25. (-5)² = 25. Say it plainly: a linear equation asks "which one number?" and gets one answer. A squared variable hides its sign - squaring erases it - so the unsquared answer comes in a ± pair. Forgetting the negative solution is the most common error in this unit, and substitution is how you catch it.

Check yourself — explain it

x² = 25 has two solutions, but x + 2 = 25 has only one, because...

Paired problem — example still visible

Solve x² = 49 and verify both solutions -> x = 7 or x = -7 (x = ±7); check: 7² = 49 and (-7)² = 49

Faded completion — last step is yours

"x² = 81: both solutions are x = ± ___" -> 9

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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