Let us recall the following concepts:
The square root of a polynomial:
The square root of a polynomial \(p(x)\) is defined as an expression \(q(x)\), when multiplied with itself, results in the given number \(p(x)\).
That is, \(p(x) = q(x) \times q(x)\)
The square root of a polynomial can be determined using three methods.
They are:
1. Factorization method
2. Completing the square method
3. Long division method
Quadratic polynomial:
The polynomial is an expression of degree \(n\) for the variable \(x\) is of the form \(p(x) = a_0 x^n + a_{1}x^{n-1} + … + a_{n -1}x + a_n\), where \(a_0, a_1, a_2, …, a_n\) are coefficients and \(a_0 \ne 0\).
A quadratic polynomial is a polynomial of degree \(2\) for the variable \(x\) is of the form \(p(x) = ax^2 + bx + c\), \(a \ne 0\) and \(a\), \(b\) and \(c\) are real numbers.
Zero of a quadratic polynomial:
If the value of the polynomial \(p(x)\) at \(x = k\) is zero \(p(k) = 0\), then the real number \(k\) is called the zero of the polynomial \(p(x)\).
Quadratic equation:
A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are numbers, \(a \ne 0\). The degree of the quadratic equation is \(2\).
Roots of a quadratic equation:
The value of \(x\) that makes the expression \(ax^2 + bx + c\) is zero, called the roots of the quadratic equation.
Formation of a quadratic equation:
If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, then the general formula to construct the quadratic equation is \(x^2 - (\alpha + \beta) x + \alpha \beta = 0\).
That is, \( x^2 - (\text{sum of roots}) x + \text{product of roots} = 0\).
Sum of the roots \(=\) \(\alpha + \beta\) \(=\) \(\frac{-b}{a}\)
Product of the roots \(=\) \(\alpha \beta\) \(=\) \(\frac{c}{a}\)
The solution of the quadratic equation is the value of the variable that makes the equation zero. We can say these solutions as roots or zeroes.
Solve by factorisation method:
Let us see the procedure to solve the quadratic equation by the factorisation method.Step 1: Write the given equation in standard form.
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
Step 3: Group the expression into two linear factors by taking the common term outside.
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
Step 3: Group the expression into two linear factors by taking the common term outside.
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Relation between roots and coefficients of a quadratic equation:
If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\), then:
(i) Sum of the roots \(=\)
(ii) Product of the roots \(=\)
Procedure to find the solution of quadratic equation graphically:
The steps to find the solution of the quadratic equation graphically is given by:
Step 1: Draw the graph of the equation \(y = ax^2 + bx + c\).
Step 2: (i) If the graph intersects the \(X\) - axis at only one point, the given quadratic equation has only one root or two real and equal roots.
(ii) If the graph intersects the \(X\) - axis at two distinct points, the given quadratic equation has two real and unequal roots.
(iii) If the graph does not intersect the \(X\) - axis at any point, the given equation has no real root.
Step 2: (i) If the graph intersects the \(X\) - axis at only one point, the given quadratic equation has only one root or two real and equal roots.
(ii) If the graph intersects the \(X\) - axis at two distinct points, the given quadratic equation has two real and unequal roots.
(iii) If the graph does not intersect the \(X\) - axis at any point, the given equation has no real root.