A quadratic equation can be solved in several forms, and the strongest method is the one that exposes the structure with the least unnecessary work. Factorisation is fast when factors are visible. Completing the square reveals the centre and extreme value of a quadratic. The quadratic formula works systematically. Graphs make roots visible as intersections.
This forty-first Secondary 4 Mathematics Learning Guide develops these methods as one decision system. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
It deepens the earlier Quadratic Equations and Algebraic Fractions and Quadratic Functions: Forms, Roots, Turning Points and Symmetry guides.
Four routes to the same quadratic relationship
- Factorisation exposes roots directly.
- Completing the square exposes a translated square and hence the turning point.
- Quadratic formula gives roots systematically.
- Graphical solution reads roots or intersections from a graph.
The methods are not competitors. They are different representations of the same equation.
Route 1 | Factorisation when the structure is visible
Solve x²−7x+12=0.
x²−7x+12=(x−3)(x−4).
Therefore x=3 or x=4.
Factorisation is efficient here because two integers with product 12 and sum −7 are immediately available.
Completing the square: building a perfect square deliberately
The identity (x+a)²=x²+2ax+a² tells us how to convert a quadratic expression into a square plus or minus a constant.
For x²+6x, half the coefficient of x is 3. Therefore:
x²+6x=(x+3)²−9.
Worked Example 2 | Complete the square
Write x²−8x+5 in completed-square form.
Half of −8 is −4:
x²−8x+5=(x−4)²−16+5=(x−4)²−11.
The completed-square form immediately reveals a minimum value −11 at x=4.
Worked Example 3 | Solve by completing the square
Solve x²+4x−5=0.
Move the constant:
x²+4x=5.
Add 4 to both sides:
(x+2)²=9.
Therefore x+2=±3, giving x=1 or x=−5.
The ± is essential. Solving a square equation usually creates two possible square-root branches.
When the coefficient of x² is not 1
First factor out the coefficient from the x² and x terms before completing the square.
Worked Example 4 | Complete the square with a leading coefficient
Write 2x²+12x+7 in completed-square form.
Factor 2 from the first two terms:
2(x²+6x)+7.
Complete the square inside:
2[(x+3)²−9]+7=2(x+3)²−11.
The minimum value is −11 at x=−3.
The quadratic formula
For ax²+bx+c=0, a≠0:
x=[−b±√(b²−4ac)]/(2a).
The formula is especially useful when the quadratic does not factorise cleanly over integers.
Worked Example 5 | Formula with non-integer roots
Solve 2x²−3x−4=0.
Here a=2, b=−3, c=−4.
x=[3±√(9+32)]/4=[3±√41]/4.
So the exact roots are x=(3+√41)/4 and x=(3−√41)/4.
The expression under the square root predicts the root pattern
The quantity b²−4ac is the discriminant.
- If b²−4ac>0, there are two distinct real roots.
- If b²−4ac=0, there is one repeated real root.
- If b²−4ac<0, there are no real roots.
This same information appears graphically as two x-axis intersections, one tangent touch, or no x-axis intersection.
Worked Example 6 | Predict without fully solving
Determine the number of real roots of 3x²+2x+5=0.
b²−4ac=2²−4(3)(5)=4−60=−56.
The discriminant is negative, so there are no real roots.
Graphical solutions are intersections
To solve f(x)=0 graphically, read the x-coordinates where y=f(x) meets the x-axis.
To solve f(x)=g(x), find where the graphs y=f(x) and y=g(x) intersect.
Worked Example 7 | Solve from an intersection condition
Find where y=x²−2x−3 meets y=1.
Set:
x²−2x−3=1.
So x²−2x−4=0 and:
x=1±√5.
A graph would show the same two intersection x-values approximately.
Graph accuracy depends on scale and reading
A graphical root is usually approximate unless the exact coordinate is obvious. Read the scale carefully and do not report more precision than the graph supports.
Graphical methods are particularly useful for checking algebraic roots or solving equations for which an exact symbolic route is not required.
Method selection under examination conditions
| Structure | Likely efficient route |
|---|---|
| Simple integer factors visible | Factorisation |
| Need turning point or minimum/maximum form | Completing the square |
| Awkward coefficients or irrational roots | Quadratic formula |
| Graph supplied or approximate root requested | Graphical solution |
| Need verification | Use a second representation |
Worked Example 8 | Choose the shortest route
Solve x²−11x+28=0.
The numbers 4 and 7 multiply to 28 and add to 11, so:
(x−4)(x−7)=0.
x=4 or x=7. Using the full quadratic formula would be correct but unnecessarily long.
Worked Example 9 | Use completed-square form to read a range
For y=3(x−2)²−12, state the minimum y-value.
Since (x−2)²≥0:
y≥−12.
The minimum occurs at x=2.
Worked Example 10 | Verify formula roots by substitution
Suppose solving x²−5x+6=0 gives x=2 and x=3. Substitute:
- x=2: 4−10+6=0.
- x=3: 9−15+6=0.
Both roots satisfy the original equation.
Common failure modes
| Error | Cause | Repair |
|---|---|---|
| Completes x²+6x as (x+3)² | Added 9 not balanced | Write (x+3)²−9 |
| Forgets ± after square root | Two branches collapsed | Write both square-root possibilities |
| Uses wrong sign for b in formula | Coefficient not identified carefully | Write a, b, c before substitution |
| Denominator written 2 instead of 2a | Formula incompletely remembered | Group entire denominator as 2a |
| Graphical root read from y-axis | Root and intercept confused | Root occurs where y=0 |
| Over-rounds exact radical root | Calculator display treated as exact | Keep radical form unless approximation requested |
Independent practice
- Write x²+10x+7 in completed-square form.
- Solve x²+6x−7=0 by completing the square.
- Solve 3x²+5x−2=0.
- Determine the number of real roots of 2x²+4x+7=0 without solving fully.
- Find where y=x²−4x+1 meets y=−2.
- State the minimum value of y=2(x+1)²−9.
Explained answers
1. x²+10x+7=(x+5)²−25+7=(x+5)²−18.
2. x²+6x=7; add 9: (x+3)²=16; x+3=±4; x=1 or −7.
3. 3x²+5x−2=(3x−1)(x+2), so x=1/3 or −2.
4. Discriminant=16−56=−40, so no real roots.
5. x²−4x+1=−2 gives x²−4x+3=0=(x−1)(x−3), so x=1 or 3.
6. Minimum value=−9.
Final thought
Quadratic fluency is not memorising four disconnected methods. It is recognising what each form exposes and choosing the representation that makes the next step easiest to see and verify.
Factor when the roots are visible, complete the square when structure matters, use the formula when the coefficients resist, and use graphs to see the equation geometrically.
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