Quadratic Equations of the Form ax squared = c

Every equation you've solved so far had one answer. Squared equations are different, and the reason is simple: squaring destroys the sign. Ask 'what number times itself gives 9?' and there are two honest answers — 3 times 3 is 9, and negative 3 times negative 3 is also 9, because a negative times a negative is positive. So x² = 9 has two solutions, x = 3 and x = −3, written compactly as x = ±3. Losing the negative solution is the single most common mistake in this topic, and it costs half the marks on every question.

The method is the same balance-scale idea you already use, with one new final move. Take 3x² = 48. First get x² on its own by dividing both sides by 3, giving x² = 16. Only then take the square root of both sides, giving x = ±4. The order matters: you cannot square-root while the 3 is still attached, because √(3x²) is not 3x. And always check both answers in the original equation — 3 × 4² = 48 and 3 × (−4)² = 48, so both are genuine solutions.

Sometimes the number isn't a perfect square. If x² = 20, then x = ±√20, which is about ±4.472. Which form you give depends on the wording: 'exact value' means leave it as ±√20, while 'to one decimal place' means ±4.5. Knowing the perfect squares by heart — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 — tells you instantly which kind of answer to expect. And one case has no answer at all: x² = −16 is impossible, because squaring any number, positive or negative, never produces a negative result.

These equations turn up whenever a square is involved. A square patio with area 49 square metres gives x² = 49, so x = ±7. Here's the important bit: you still solve for both answers, but then you reject x = −7 because a length can't be negative, and you say so in your working — that reasoning is worth marks. It also matters that the rejection isn't automatic. Lengths, areas, times and counts can't be negative, but temperatures, money owed and positions on a number line certainly can, so check what the variable actually represents before discarding anything.

The full checklist. One: isolate x² using division and the usual moves. Two: square root both sides. Three: write ±, because there are two answers. Four: decide whether the question wants an exact surd or a rounded decimal. Five: if it's a real-world problem, reject the impossible answer and give the reason. Six: check both answers in the original equation. Step three is where half the marks in this topic live — students who do everything else perfectly and write x = 4 instead of x = ±4 lose them every single time.