Formation of Quadratic Equation
Forming a quadratic expression
We know how to find roots when we know the quadratic expression. How about the converse: if we know roots, can we find the corresponding quadratic expression? The problem is that for the given roots, the corresponding expression is not unique. Actually for given roots, there are infinte expressions possible which have the same roots. For examples, \(2(x-5)(x+2)\) and \(8(x-5)(x+2)\) have the same roots \(5\) and \(-2\). In general, any quadratic expression \(a(x-5)(x+2)\), where \(a\in \mathbb R-\{0\}\) (\(a\) is any real number expcept \(0\)) has roots \(5\) and \(-2\).
So, if somebody asks you to find a quadratic expression with roots \(\alpha\) and \(\beta\), then you would say, that there is not 'a' quadratic expression with these roots but there are infinitely many, given by, \(a(x-\alpha)(x-\beta)\), where \(a\) is a non-zero real number. If you are given more information about the quadratic expression, it might then be possible to find the exact value of \(a\).
For example, find a quadratic expression whose roots are \(-1\) and \(3\). You are given some additional information: the graph of the expression passes through \((2, 4)\).
Solution: Given the roots are \(-1\) and \(3\), the quadratic expression can be assumed to be in the form: \( a(x + 1)(x - 3) \)
Now, we'll use the fact that the graph of the expression passes through the point \((2, 4)\). This means when \(x = 2\), the value of the expression is \(4\). We can use this information to find the value of \(a\).
Put \(x=2\) and put the value of expression equalto \(4\). We get the equation: \( 4 = a(2 + 1)(2 - 3) \)
Let's calculate this to find \(a\).
The value of \(a\) is \(-4\). Therefore, the specific quadratic expression with roots \(-1\) and \(3\), and that passes through the point \((2, 4)\), is given by: \( -4(x + 1)(x - 3) \)
Expanding this expression, we get: \( -4(x^2 - 2x - 3) = -4x^2 + 8x + 12 \)
Hence, the quadratic expression is: \( -4x^2 + 8x + 12 \)
Forming a quadratic expression with sum and product of roots
To form a quadratic expression, we need not know the exact values of \(\alpha\) and \(\beta\), we need to know sum and product of roots. This is because a quadratic expression whose roots are \(\alpha\) and \(\beta\) is \(a(x-\alpha)(x-\beta)\).
Starting with \(a(x-\alpha)(x-\beta)\):
- Expand the expression: \( a(x-\alpha)(x-\beta) = a(x^2 - \beta x - \alpha x + \alpha \beta) \)
- Simplify the expression: \( = a(x^2 - (\alpha + \beta)x + \alpha \beta) \)
This says that knowing the sum and product of the roots \(\alpha\) and \(\beta\) is sufficient to form the quadratic expression.
When you know the sum and product of its roots, denoted as \(S\) and \(P\) respectively, the quadratic expression can be written as:
Here, \(a\) is a non-zero constant that can be any real number, whose value can be decided by some additional information.
This format is very useful for quickly constructing a quadratic expression when you know the sum and product of its roots.
Forming a quadratic equation
The corresponing equation whose roots are \(\alpha\) and \(\beta\) is given by \((x-\alpha)(x-\beta)=0\). There is no requirement of putting \(a\), that is, \(a(x-\alpha)(x-\beta)=0\) \(\implies(x-\alpha)(x-\beta)=0\). Whatever be the value of \(a\), the equation is always unique. In other words, when asked to write a quadratic equation with given roots, we write the simplext equation with \(a=1\).
Similarly, the quadratic equation, in the simplest form, whose sun of roots is \(S\) and product of roots is \(P\), is given by, \(x^2-Sx+P=0\).