Learning Objectives
  • Recognise quadratic expressions and equations in standard form.
  • Solve quadratic equations by factorisation where suitable.
  • Construct tables of values and draw accurate quadratic graphs.
  • Identify roots, intercepts, the axis of symmetry and the turning point.
  • Use completed-square form to sketch and interpret a quadratic graph.
Key Terms
Quadratic
An expression or equation whose highest power of the variable is 2.
Root
A value of x for which an equation equals zero; graphically, an x-intercept.
Parabola
The U-shaped or inverted U-shaped graph of a quadratic function.
Axis of symmetry
The vertical line that divides a parabola into mirror-image halves.
Turning point
The maximum or minimum point of a quadratic graph.
Completed-square form
A form such as a(x-h)^2+k that reveals the turning point.
Y-intercept
The value of y when x=0.
What This Chapter Covers
  • Quadratic equations in standard form
  • Solving by factorisation
  • Plotting quadratic graphs from tables
  • Roots, intercepts, symmetry and turning points
  • Completing the square for graph interpretation
Recognising A Quadratic

A quadratic equation can be written in the form ax^2+bx+c=0, where a\ne0. The graph of y=ax^2+bx+c is a parabola. If a>0, it opens upwards and has a minimum. If a<0, it opens downwards and has a maximum.

ax^2+bx+c=0\qquad(a\ne0)
Solving By Factorisation

The zero-product rule states that if AB=0, then A=0 or B=0. Therefore, once a quadratic has been rearranged to zero and factorised, set each factor equal to zero.

Factorisation Method
  1. Rearrange the equation so that one side is zero.
  2. Factorise the quadratic fully.
  3. Apply the zero-product rule to each linear factor.
  4. Solve the resulting linear equations.
  5. Check the solutions in the original equation if the algebra was complicated.
Worked Example: Monic Quadratic

Question: Solve x^2-7x+12=0.

  1. Find two numbers with product 12 and sum -7: -3 and -4.
  2. Factorise: (x-3)(x-4)=0.
  3. Set each factor equal to zero.

Answer: x=3 or x=4.

Worked Example: Leading Coefficient Not One

Question: Solve 2x^2+x-6=0.

  1. Factorise by splitting the middle term: 2x^2+4x-3x-6=0.
  2. Group: 2x(x+2)-3(x+2)=0.
  3. Factorise: (2x-3)(x+2)=0.

Answer: x=\frac32 or x=-2.

Constructing A Table Of Values

Choose a suitable interval that captures the important features of the graph. Substitute each x-value carefully. Negative values must be bracketed before squaring. If the graph is symmetric, calculated values should reflect that symmetry; this provides a useful check.

Worked Example: Table For A Quadratic

Question: For y=x^2-4x-5, calculate y when x=-1,0,1,2,3,4,5.

  1. At x=-1, y=1+4-5=0.
  2. At x=0, y=-5.
  3. At x=1, y=-8.
  4. At x=2, y=-9.
  5. By symmetry about x=2, the values at 3, 4 and 5 are -8,-5,0.

Answer: The points show a minimum at (2,-9) and roots -1 and 5.

Plotting The Graph

Use an even scale, plot points as small crosses and draw a smooth curve. Do not join the points with separate straight segments. Extend the curve only as far as the scale allows. Label both axes and mark key values clearly.

Roots And Intercepts

The roots are the x-coordinates where the graph crosses or touches the x-axis, because y=0 there. The y-intercept is found by setting x=0, so for y=ax^2+bx+c, the y-intercept is c. A graph may have two distinct real roots, one repeated real root where it just touches the axis, or no real roots.

Feature How To Find It Algebraically Meaning On The Graph
Roots Solve ax^2+bx+c=0 Where the curve meets the x-axis
Y-intercept Set x=0 Where the curve meets the y-axis
Axis of symmetry Midway between the roots or from completed-square form Vertical mirror line
Turning point Use completed square or substitute axis value Maximum or minimum point
Completing The Square

Completed-square form reveals the turning point immediately. For a monic quadratic, take half the coefficient of x, square it, and compensate for the added square.

Worked Example: Complete The Square

Question: Write x^2-6x+2 in completed-square form.

  1. Half of -6 is -3.
  2. (x-3)^2=x^2-6x+9.
  3. Subtract 9 and add the original constant 2.

Answer: x^2-6x+2=(x-3)^2-7.

Therefore the graph y=(x-3)^2-7 has axis of symmetry x=3 and minimum point (3,-7).

Worked Example: Negative Leading Coefficient

Question: Find the turning point of y=-2x^2+8x-3.

  1. Factor -2 from the first two terms: y=-2(x^2-4x)-3.
  2. Complete the square: x^2-4x=(x-2)^2-4.
  3. Substitute: y=-2[(x-2)^2-4]-3=-2(x-2)^2+5.

Answer: The turning point is (2,5), and it is a maximum.

Sketching Without A Full Table

A useful sketch must show the correct opening direction, x-intercepts if real, y-intercept, turning point and axis of symmetry. It need not be perfectly to scale, but all labelled features must be consistent.

Quick Sketch Checklist
  1. Check the sign of a to determine whether the parabola opens up or down.
  2. Find the y-intercept c.
  3. Find roots by factorisation where possible.
  4. Find the axis of symmetry midway between the roots or by completing the square.
  5. Find the turning point and label it.
  6. Draw a smooth symmetric curve through the features.
Using A Quadratic Graph To Solve An Equation

If the graph of y=f(x) is drawn, solutions of f(x)=0 are read from the x-axis. To solve f(x)=k, draw the horizontal line y=k and read the x-coordinates of intersections. Read values only to the accuracy supported by the graph scale.

Worked Example: Graphical Interpretation

Question: The graph of y=x^2-4x-5 is drawn. Explain how to solve x^2-4x-5=3.

  1. Draw the horizontal line y=3.
  2. Locate its two intersections with the parabola.
  3. Read the corresponding x-coordinates from the horizontal axis.

Answer: The intersection x-coordinates are the approximate solutions.

Quadratics In Context

Quadratic relationships can model areas, projectile paths and optimisation situations. A context may restrict the acceptable root. For example, if a solution represents a length or time, a negative root may be mathematically valid but physically impossible.

Examination Guidance
  • Rearrange to zero before factorising a quadratic equation.
  • For graph questions, show a clear scale and plot accurately enough to justify readings.
  • Use completed-square form to state the turning point directly.
  • When a question asks for a sketch, label intercepts, turning point and axes rather than producing an unlabelled curve.
  • Keep exact roots if the question does not ask for decimals.
Common Mistakes
  • Forgetting the second root after setting one factor equal to zero.
  • Squaring a negative input without brackets when making a table.
  • Joining plotted points with straight line segments instead of a smooth curve.
  • Confusing the turning point x-coordinate with the minimum or maximum y-value.
  • Reading roots from the y-axis instead of the x-axis.
Knowledge Check And Practice

1. Solve x^2-9x+20=0.

Answer: (x-4)(x-5)=0, so x=4 or 5.

2. Solve 3x^2-5x-2=0 by factorisation.

Answer: (3x+1)(x-2)=0, so x=-1/3 or 2.

3. State the y-intercept of y=2x^2-7x+6.

Answer: 6.

4. Write x^2+8x+3 in completed-square form.

Answer: (x+4)^2-13.

5. State the turning point of y=(x-5)^2+2.

Answer: (5,2), a minimum.

6. Find the axis of symmetry of a quadratic with roots -2 and 8.

Answer: x=3.

7. A quadratic opens downwards. What can be said about its turning point?

Answer: It is a maximum point.

8. How would you use the graph of y=x^2+x-6 to solve x^2+x-6=4?

Answer: Draw y=4 and read the x-coordinates of the intersections.
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