Quadratic Expressions and Equations (II)
Algebra · Quadratic Expressions and Equations (II)
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 1 Quadratic Expressions and Equations (II)
Lesson objectives
By the end of this topic, you should be able to:
- Factorise quadratic expressions and identify perfect squares.
- Solve quadratic equations by completing the square.
- Derive and apply the quadratic formula.
- Form quadratic equations from given roots and solve graphically.
Quadratic Expressions and Equations (II)
Form 2 solved quadratics by factorising. That only works when the expression factorises neatly. This topic handles every quadratic.
a) Perfect squares
A perfect square trinomial comes from squaring a bracket.
So x² + 6x + 9 is (x + 3)². And x² − 10x + 25 is (x − 5)².
The pattern is a² ± 2ab + b². The constant is always the square of half the middle coefficient.
Half of 6 is 3, and 3² is 9. That relationship is the key to the next method.
b) Completing the square
Move the constant to the right.
Take half the coefficient of x, square it, and add it to both sides.
The left side is now a perfect square. Factorise it.
Take the square root of both sides, remembering ±.
Solve the two resulting equations.
Forgetting the ± loses one of the two roots, and that is the commonest error here.
c) When a is not 1
Divide every term by a first, so the x² coefficient becomes 1.
Take 2x² + 8x + 6 = 0. Divide by 2 to get x² + 4x + 3 = 0, then proceed as before.
d) The quadratic formula
The formula comes from completing the square on ax² + bx + c = 0 in general. It is not a separate method but the same one, done once for all cases.
Substitute a, b and c carefully, keeping their signs.
Write the formula down before substituting. Most errors come from sign slips in −b or in b² − 4ac.
e) The discriminant
The quantity b² − 4ac is the discriminant.
If it is positive, there are two distinct real roots.
If it is zero, there is one repeated root, and the expression is a perfect square.
If it is negative, there are no real roots. The square root of a negative is not real.
Calculating the discriminant first tells you what to expect before doing the full working.
f) Forming an equation from its roots
If the roots are p and q, the equation is (x − p)(x − q) = 0.
Expanding gives x² − (p + q)x + pq = 0.
So the equation is x² − (sum of roots)x + (product of roots) = 0.
Take roots 3 and −5. The sum is −2 and the product is −15. So the equation is x² + 2x − 15 = 0.
g) Solving graphically
Draw the graph of y = ax² + bx + c.
The roots are where the curve crosses the x-axis, because y is zero there.
Two crossings mean two roots, one touch means a repeated root, and no crossing means no real roots. That matches the discriminant exactly.
To solve a different equation from the same graph, rearrange it so one side matches the drawn curve. Draw the line given by the other side, then read the intersections.
h) Where this is used
Projectile paths. Areas where a dimension is unknown. Optimisation problems. Any relationship where a quantity multiplies by itself.
Words to know
- Perfect square trinomial -- an expression factorising as a single bracket squared.
- Completing the square -- rewriting a quadratic as a squared bracket plus a constant.
- Discriminant -- the quantity b² − 4ac, determining the number of real roots.
- Repeated root -- the single root occurring when the discriminant is zero.
- Turning point -- the maximum or minimum point of a parabola.
:::checkpoint Check yourself
- Solve x² + 8x + 7 = 0 by completing the square.
- Solve 2x² − 5x − 3 = 0 using the formula.
- What does a discriminant of zero tell you?
- Form the quadratic equation whose roots are 4 and −2. :::
Bridge to practice
The exercises begin with perfect squares and completing the square, move through the formula and the discriminant, and finish with forming equations from roots and graphical solutions. Whenever you take a square root of both sides, write the ± immediately, before doing anything else.