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Quadratic Equations

Mathematics · WAEC and JAMB · SS2 and SS3

A quadratic appears in Paper 2 almost without fail, often as a graph question worth ten or more marks. The three methods are not interchangeable in the exam: the paper usually names the one it wants, and using another method scores nothing.

What you need to know

  • A quadratic equation has the form ax^2 + bx + c = 0 with a not equal to zero, and it has at most two roots. Before you solve anything, rearrange so that one side is zero.
  • To solve by factorising, split the middle term using two numbers whose product is a x c and whose sum is b, group, then set each bracket to zero. Factorisation only works cleanly when the roots are rational.
  • The zero product rule is the reason factorising works: if two quantities multiply to give zero, at least one of them must be zero. Never divide both sides by x, because you lose the root x = 0.
  • To complete the square, make the coefficient of x^2 equal to 1, move the constant to the right, then add the square of half the coefficient of x to BOTH sides.
  • Completing the square also gives the turning point: writing y = (x - h)^2 + k shows the minimum point at (h, k) with no graph needed.
  • The quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a) always works. Substitute with brackets around negative values, since the commonest slip is losing the sign of b.
  • The discriminant b^2 - 4ac tells you the nature of the roots before you solve: positive means two distinct real roots, zero means one repeated root, negative means no real roots.
  • For any quadratic, the sum of the roots is -b/a and the product is c/a. To build an equation from roots, use x^2 - (sum)x + (product) = 0.
  • Given roots 2 and -5, the sum is -3 and the product is -10, so the equation is x^2 + 3x - 10 = 0. Watch the sign change on the sum term.
  • The graph of a quadratic is a parabola. It opens upward when a is positive and downward when a is negative, which tells you at once whether the turning point is a minimum or a maximum.
  • The roots of the equation are where the curve CROSSES the x-axis, that is where y = 0. If the curve does not touch the axis, there are no real roots.
  • Graph questions require a clearly stated scale, a full table of values, plotted points and a smooth freehand curve, not straight segments between points. Each of these carries marks.
  • To solve a different equation from the same graph, rearrange it so that one side matches the plotted function and draw the straight line given by the other side, then read off the x-coordinates of the intersections.
  • Always give roots to the accuracy the paper demands, usually two decimal places or one decimal place for graphical solutions, and keep both roots unless the context rules one out, as with a negative length.

Key terms

Quadratic equation
An equation of the form ax^2 + bx + c = 0 in which the highest power of the unknown is two and a is not zero.
Root
A value of the unknown that makes the equation true, shown on the graph as a point where the curve meets the x-axis.
Discriminant
The expression b^2 - 4ac, whose sign determines the number and nature of the roots.
Completing the square
Rewriting a quadratic in the form (x + p)^2 + q so that the unknown appears only once.
Parabola
The U-shaped curve produced by a quadratic function, symmetrical about a vertical line through its turning point.
Turning point
The maximum or minimum point of a parabola, where the curve changes direction.

Formulae

  • general form: ax^2 + bx + c = 0, a not equal to 0
  • quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
  • discriminant: D = b^2 - 4ac
  • sum of roots = -b/a
  • product of roots = c/a
  • equation from roots: x^2 - (sum of roots)x + (product of roots) = 0
  • line of symmetry: x = -b / (2a)

Worked examples

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

  1. Identify a = 2, b = -5, c = -3, so a x c = 2 x -3 = -6.
  2. Find two numbers with product -6 and sum -5. They are -6 and +1.
  3. Split the middle term: 2x^2 - 6x + x - 3 = 0.
  4. Group: (2x^2 - 6x) + (x - 3) = 0, which gives 2x(x - 3) + 1(x - 3) = 0.
  5. Factor out the common bracket: (2x + 1)(x - 3) = 0.
  6. Set each factor to zero: 2x + 1 = 0 gives x = -1/2, and x - 3 = 0 gives x = 3.
  7. Check x = 3: 2(9) - 15 - 3 = 18 - 18 = 0. Check x = -1/2: 2(0.25) + 2.5 - 3 = 0.5 + 2.5 - 3 = 0.

Solve x^2 - 6x + 4 = 0 by completing the square, giving your answers correct to two decimal places.

  1. The coefficient of x^2 is already 1. Move the constant to the right: x^2 - 6x = -4.
  2. Half the coefficient of x is -3, and its square is 9. Add 9 to both sides: x^2 - 6x + 9 = -4 + 9.
  3. The left side is now a perfect square: (x - 3)^2 = 5.
  4. Take the square root of both sides, remembering both signs: x - 3 = +/- sqrt(5).
  5. Since sqrt(5) = 2.236, x = 3 + 2.236 = 5.236 or x = 3 - 2.236 = 0.764.
  6. Round to two decimal places: x = 5.24 or x = 0.76.
  7. Check with the sum of roots: 5.236 + 0.764 = 6, which equals -b/a = 6.

Find the quadratic equation whose roots are 2 and -5.

  1. Sum of roots = 2 + (-5) = -3.
  2. Product of roots = 2 x (-5) = -10.
  3. Use x^2 - (sum)x + (product) = 0.
  4. Substitute: x^2 - (-3)x + (-10) = 0.
  5. Simplify: x^2 + 3x - 10 = 0.
  6. Check by factorising: (x + 5)(x - 2) = 0 gives x = -5 or x = 2.

The mistake to avoid

Candidates solve by factorising when the question says 'using the quadratic formula' or 'by completing the square', and the examiner awards nothing because the method itself is what is being marked. The second error is dropping the negative root, especially the plus-or-minus after taking a square root.

In the exam

Underline the method named in the question before you write a single line, and use exactly that method even if another is faster. For graph questions, draw the table of values in full, state the scale you used, and read intersections to one decimal place; the table, the scale and the smooth curve each carry marks independent of your final answer.