A graphical solution is a value read from a graph or from the calculator’s solver, so it is an approximation. An exact solution comes from algebra and may include a fraction or a surd such as √5. Exams ask for one or the other depending on the command words, so you need to know which you are producing.
How do you decide which one the question wants?
Look at the wording before you touch the calculator.
- “Solve”, “show that”, “find the exact value”: algebra is required. A graph can guide you, but it is not the answer.
- “Use your graph to solve” or “estimate”: a reading from the graph is expected, usually with a stated accuracy.
- “Give your answer correct to 3 significant figures”: calculate exactly or with the solver, then round once at the end.
When both appear, use the graph to see how many solutions to expect and to check your algebra.
Worked example
Solve x² = x + 6 graphically, then exactly.
Graph: plot y = x² and y = x + 6. The intersection tool gives x = −2 and x = 3, with points (−2, 4) and (3, 9).
Exact: x² − x − 6 = 0, so (x − 3)(x + 2) = 0, giving x = 3 or x = −2.
Check by substitution: 3² = 9 and 3 + 6 = 9. (−2)² = 4 and −2 + 6 = 4. Both sides match.
Here the graph gives whole numbers, so graph and algebra agree completely. That does not always happen.
When the decimal is not the whole story
Solve x² − 2x − 4 = 0.
Graph: the calculator shows roots near x = 3.236067977 and x = −1.236067977.
Exact: the formula gives x = (2 ± √(4 + 16))/2 = (2 ± √20)/2 = 1 ± √5.
Check: √5 ≈ 2.236068, so 1 + √5 ≈ 3.236068 and 1 − √5 ≈ −1.236068. These match the screen. To 2 decimal places the answers are 3.24 and −1.24.
The surd form 1 ± √5 is the exact answer. The decimals are approximations of it.
The mistake to watch for
Mistaken working: “x² − 2x − 4 = 0. From my graph, x = 3.2 and x = −1.2.”
Two problems appear. The values are rounded to 1 decimal place, although the question did not say so. More importantly, if the question said “solve”, a graph reading gives a clue but not a method.
The correction is to write the equation, use the formula or completing the square, and then state the exact form and the rounded form. The graph becomes a check, not the proof.
Check yourself
1. Solve x² = 2x + 8 exactly. Then say what the intersection of y = x² and y = 2x + 8 would show.
Show answer
x² − 2x − 8 = 0 gives (x − 4)(x + 2) = 0, so x = 4 or x = −2. Check: 16 = 8 + 8 and 4 = −4 + 8. The graphs intersect at (4, 16) and (−2, 4).
2. Solve x² + 3x − 1 = 0, giving answers to 2 decimal places.
Show answer
x = (−3 ± √(9 + 4))/2 = (−3 ± √13)/2. √13 ≈ 3.605551. So x ≈ 0.302776 or x ≈ −3.302776. To 2 decimal places: x = 0.30 or x = −3.30.
3. The calculator shows a root of x² − 2 = 0 as 1.414213562. What is the exact value, and how would you give it to 3 significant figures?
Show answer
x² = 2, so the positive root is exactly √2, and the other is −√2. To 3 significant figures, √2 ≈ 1.41.
Where this leads next
Next, see how many decimal places or significant figures to keep. If you want to see roots, turning point and forms together, the quadratic structure explorer helps, and the graph-model explorer shows how a fitted graph compares with data. The module overview lists the full route.
Some students can solve by algebra and by graph but mix up which one the question wants. That is something a teacher in online one-to-one Mathematics tuition can drill with your own past mistakes.