Many IB mathematics tutors consider quadratic equations as a very important topic of ib maths. There are following ways to solve a quadratic equation
► Factorization method
►complete square method
► graphical method
► Quadratic formula method
The quadratic formula is the strongest method to solve a quadratic equation. In this article, I will use …… steps to prove the quadratic formula
Given equation: ax²+bx+c=0
Step-1: transfer constant term to right side
Step-2: divide both sides by coefficient of x²
Step-3: write (coefficient of x/2)² that is (b/2a)²=b²/4a²
Step-4: Add this value to both sides
now, take square root on both sides
This formula is known as quadratic formula, we can put values of a, b and c from any equation and find the value of x (the variable) by directly using this formula.
IB Mathematics tutors can also explain the concept of conjugate roots with the help of quadratic formula. In a quadratic equation,
if a, b and c are all rational numbers and one root of the quadratic equation is a+√b then the second root will automatically become a-√b. that can be understood easily as we use one +ve and one -ve sign in quadratic formula.
These types of roots are called Conjugate Roots.
If a & b are the roots of the quadratic equation ax² + bx + c = 0, then;
Nature Of Roots:
(a) Consider the quadratic equation ax² + bx + c = 0 where a, b, c
(i) D > 0
roots are real & distinct (unequal).
(ii) D = 0
roots are real & coincident (equal).
(iii) D < 0
roots are imaginary
(B) Consider the quadratic equation ax2+ bx + c = 0 where a, b, c Q & then
If D > 0 & is a perfect square , then roots are rational & unequal.
A quadratic equation whose roots are a & b is (x – a)(x – b) = 0 i.e. x2 – (a + b) x + a b = 0 i.e.
x2 – (sum of roots) x + product of roots = 0
Consider the quadratic expression , y = ax² + bx + c , a, b, c R & then
(i) The graph between x, y is always a parabola. If a > 0 then the shape of the parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards.
Common Roots Of 2 Quadratic Equations [Only One Common Root]-
be the common root of ax² + bx + c = 0 & a’x2 + b’x + c’ = 0
If we solve above pair by cramer’s rule we get
This will give us
Every pair of the quadratic equation whose coefficients fulfils the above condition will have one root in common.
The condition that a quadratic function-
f(x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c may be resolved into two linear factors is that
Reducible Quadratic Equations-These are the equations which are not quadratic in their initial condition but after some calculations, we can reduce them into quadratic equations
(i) If the power of the second term is exactly half to the power of the first term and the third term is a constant, these types of equations can be reduced to quadratic equations.
all these equations can easily be reduced into quadratic equations by applying the method of substitution.
Example-Solve this equation and find x
let y= then given equation will become
it’s a simple quadratic equation we can be easily factorised it and solve so y=–3,2
the final equation can be solved using Quadratic formula and the same process can be repeated for y=2
(ii) If a variable is added with its own reciprocal, then we get a quadratic equation i.e , all these equations can be reduced into quadratic by replacing one term by any other variable.
Standard Form of a Quadratic Function-A quadratic function y=ax2+ bx + c can be
reduced into standard form by method of completing the square. If we
draw the graph of this function we shall get a parabola with vertex (h,k). The parabola will be upward for a>0 and downward for a<0
Maximum and Minimum value of a quadratic function- If the function is in the form
Then ‘h’ is the input value of the function while ‘k’ is its output.
(i) If a>0 (in case of upward parabola) the minimum value of f is f(h)=k
(ii) If a<0 (in case of downward parabola)the maximum value of f is f(h)=k
If our function is in the form of y=ax² + bx + c then vertex of the parabola
The line passing through vertex and parallel to the y-axis is called the axis of symmetry.
The parabolic graph of a quadratic function is symmetrical about axis of symmetry.
In the next post about quadratics, I shall discuss discriminant, nature of roots, relationships between the roots. In the meantime, you can download the pdf and solve practice questions.