Secondary Mathematics · Worked Repair Guide 38
An equation can be solved by finding where two representations agree. Algebra does this symbolically. A graph does it geometrically: a root is where a curve meets the x-axis, and a simultaneous solution is where two graphs meet each other.
This guide develops one central habit: translate the equation into a graph-intersection question before reading coordinates. The graphical method is especially useful when exact factorisation is unavailable, when an approximate solution is acceptable, or when the number of solutions matters as much as their values.
All functions and sketches below are original teaching constructions. Use Graphs, Tables and Relationships for graph foundations and Quadratic Equations, Factorisation and Roots for symbolic quadratic methods.
1. A root is an x-value where y=0
For y=x²−5x+6, roots satisfy x²−5x+6=0.
Graphically, they are the x-coordinates where the curve crosses or touches the x-axis.
Since x²−5x+6=(x−2)(x−3), the exact roots are2 and3. A correctly drawn graph should meet the x-axis at(2,0) and(3,0).
Entry check: the root is2, while the x-intercept is the point(2,0). State the object the question requests.
2. Solving f(x)=k means intersecting y=f(x) with y=k
To solve x²=9 graphically, one can draw y=x² and the horizontal line y=9.
The intersection x-values are−3 and3.
This is different from finding roots of y=x², which would only give x=0.
The right comparison line depends on the equation target.
3. Rearrange equations into two graphable expressions
To solve x²−2x=3 graphically, several equivalent graphing choices are possible.
One route draws y=x²−2x and y=3.
Another draws y=x² and y=2x+3.
Both intersections satisfy the same original equation. The most useful arrangement depends on which graphs are already available or easiest to construct.
4. Intersections solve simultaneous equations
Suppose y=2x+1 and y=x²−2.
At an intersection, both y-values are equal, so x²−2=2x+1.
This gives x²−2x−3=0=(x−3)(x+1), so x=−1 or3.
Corresponding y-values are−1 and7. The graphical simultaneous solutions are approximately(−1,−1) and(3,7).
5. A graph can show zero, one or two real quadratic solutions
A horizontal line may miss a parabola, touch it once or cut it twice.
These cases correspond to zero, one or two real solutions of the associated equation.
For y=x², the equation x²=−1 has no real graphical intersections with y=−1.
The equation x²=0 touches once at x=0; x²=4 intersects twice at x=±2.
6. Tangency corresponds to a repeated graphical solution
For y=(x−2)², the x-axis touches the parabola at(2,0) without crossing it.
The equation (x−2)²=0 has the repeated root x=2.
Graphically, one point of contact can represent a double algebraic root.
This is a useful bridge to discriminant and tangency ideas in later algebra.
7. Graphical solutions are usually approximate unless the scale gives exact values
If a curve crosses the x-axis near x=1.37, the graph may support x≈1.37 to the precision justified by its scale.
Do not report six decimal places from a graph whose grid can only distinguish tenths.
Graph-reading precision is limited by line thickness, scale and plotting accuracy.
Use the approximation sign where appropriate.
8. A table can locate an interval containing a root
Suppose f(1)=−0.4 and f(1.5)=0.7, and the graph is continuous between those x-values.
The sign change shows the graph crosses the x-axis somewhere between1 and1.5.
A finer table or graph can narrow the interval.
This is not yet an exact algebraic solution; it is a bracket for an approximate root.
9. Interpolation from a graph must respect the drawn curve
If a plotted curve passes through values near(2,3) and(3,5), a target y=4 may be read from the graph.
But assuming the answer is exactly x=2.5 is justified only if the segment is actually linear or the graph indicates that value.
Curved graphs do not support ordinary linear interpolation without an additional approximation assumption.
Read the curve, not just the neighbouring endpoints.
10. Solve a reciprocal-line equation by intersections
Consider 6/x=x+1 with x≠0.
Draw y=6/x and y=x+1. Their intersections solve the equation.
Algebraically, 6=x²+x, so x²+x−6=0 and x=2 or−3.
The graph should therefore show intersections at(2,3) and(−3,−2).
11. Graphs can solve equations that are awkward to rearrange exactly
Consider 2ˣ=x+3.
A graph of y=2ˣ and y=x+3 can reveal approximate intersections even when elementary algebra does not isolate x conveniently.
Some intersections may be exact by inspection: x=1 gives2=4? No. x=2 gives4=5? No. x=3 gives8=6? No. The graph then becomes useful for locating non-obvious solutions or confirming none in a range.
Graphical solving is a legitimate numerical method when exact symbolic manipulation is unavailable or unnecessary.
12. Check an approximate root by substitution
If a graph suggests x≈1.41 solves x²=2, substitute:1.41²=1.9881, close to2.
Using x≈1.414 gives1.999396, closer still.
Substitution checks whether the read-off value is plausible to the claimed precision.
Do not treat a graph reading as self-verifying when a simple numerical check is available.
13. Graphical inequalities use regions above or below a comparison line
To solve x²<4 graphically, compare y=x² with y=4.
The curves meet at x=−2 and2.
Between those values, the parabola lies below y=4.
Therefore −2<x<2.
14. Equality determines endpoint inclusion
For x²≤4, the intersection points are included, giving−2≤x≤2.
For x²>4, the solution lies outside the intersections: x<−2 or x>2.
The graph determines where one function is above another; the inequality symbol determines whether boundary intersections belong.
This connects to the existing inequalities and interval guide.
15. A graphical solution must preserve domain restrictions
For 1/(x−2)=3, x=2 is forbidden before any graph is drawn.
The graph y=1/(x−2) has a vertical asymptote at x=2 and intersects y=3 at x=7/3.
A plotting artefact or badly joined curve near x=2 must not create a false solution.
Domain restrictions remain active in graphical methods.
16. Number of intersections can answer a parameter question
Consider y=x² and horizontal line y=k.
If k<0, there are no real intersections. If k=0, one touching intersection. If k>0, two intersections.
Thus the graph classifies how the number of real solutions of x²=k changes with parameter k.
This is a visual version of threshold reasoning.
17. Transforming an equation can make graphing easier
To solve x³−x=2, one could draw y=x³−x and y=2.
Alternatively draw y=x³ and y=x+2.
Both are mathematically equivalent intersection tasks.
Choose the version that reuses an available graph or gives simpler plotting values.
18. Capstone: roots, intersections and inequalities from one parabola
Let f(x)=x²−4x+3=(x−1)(x−3).
Roots: x=1 and3.
To solve f(x)=−1, intersect the parabola with y=−1. Algebraically x²−4x+4=0, so x=2; the horizontal line touches at the vertex(2,−1).
To solve f(x)<0, identify where the parabola lies below the x-axis:1<x<3.
One graph therefore encodes roots, a tangency threshold and an inequality interval.
19. Independent practice
- For y=x²−5x+6, state the graphical roots.
- State the corresponding x-intercepts.
- How would you solve x²=9 using y=x²?
- For y=2x+1 and y=x²−2, find the intersection x-values.
- Find the corresponding y-values in Question4.
- How many real solutions does x²=−1 have?
- How many does x²=0 have?
- How many does x²=4 have?
- Explain why a tangent intersection can represent a repeated root.
- If f(1)<0 and f(1.5)>0 for a continuous graph, what can be inferred?
- Solve6/x=x+1 exactly to check a graphical method.
- Check whether x=1.41 is a reasonable approximation to √2.
- Solve x²<4 graphically.
- Solve x²≤4 graphically.
- Solve x²>4 graphically.
- State the forbidden x-value in1/(x−2)=3.
- Solve1/(x−2)=3.
- For y=x² and y=k, state the number of intersections when k<0,k=0,k>0.
- For f(x)=x²−4x+3, solve f(x)<0.
- For the same f, solve f(x)=−1.
20. Worked answers
1. x=2,3.
2. (2,0) and(3,0).
3. Intersect y=x² with y=9; x=−3,3.
4. x=−1,3.
5. y=−1,7.
6. Zero.
7. One.
8. Two.
9. The line and curve meet at one point without crossing; algebraically the corresponding equation has a repeated factor.
10. At least one root lies between1 and1.5, assuming continuity.
11. x=2 or−3.
12. Yes. 1.41²=1.9881, close to2.
13. −2<x<2.
14. −2≤x≤2.
15. x<−2 or x>2.
16. x=2.
17. x=7/3.
18. 0,1,2 intersections respectively.
19. 1<x<3.
20. x=2.
21. Diagnose graphical-solution errors by translating the equation first
Common failures include reading y-values instead of x-values, intersecting with the wrong horizontal line, forgetting a domain restriction, reporting excessive precision, or treating a visual near-intersection as exact equality.
A useful repair note says “equation means equal y-values”, “root means y=0”, “read x at the intersection”, or “graph gives approximate precision”.
Then rearrange the same equation into a different pair of graphable expressions. The solution set should remain unchanged if the rearrangement is valid.
22. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Functions, Domain, Range and Graph Behaviour, Quadratic Equations, Factorisation and Roots and Simultaneous Equations, Elimination and Modelling for neighbouring methods.
Within Batch10, continue to Radian Measure, Arc Length, Sector and Segment Area, Three-Dimensional Trigonometry and Spatial Diagrams, or Plane Mensuration: Parallelograms, Trapezia and Composite Figures.
23. Sources and scope
The functions, intersections and practice questions are original teaching material. Graphical solution statements use standard coordinate-graph interpretation and ordinary continuity assumptions where stated.
For current Singapore Secondary curriculum and assessment scope, consult the relevant MOE/SEAB syllabus for the learner’s subject level and year. Match numerical graph-solving precision and nonlinear extensions to the actual course being studied.
Complete Secondary Mathematics repair route: browse all 48 worked repair guides or return to the Secondary Mathematics Learning Hub.