A polynomial is an expression consisting of variables and constants combined using addition, subtraction, and multiplication operations, with non-negative integer powers of the variables.
For example, consider the polynomial \(p(x)=x^2+3x+2\).
This is a polynomial in the variable \(x\).
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 is a polynomial equation of degree two.
The general or standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are real numbers and \(a≠0\).
The values of the variable that satisfy the equation are called the solutions, roots, or zeroes of the quadratic equation.
In general, if \(\alpha\) is a root of the quadratic equation \(ax^2 + bx + c = 0\) then \(a \alpha^2 + b \alpha + c = 0\).
Which means \(x = \alpha\) is a solution of the quadratic equation.
Methods of solving quadratic equations:
There are three methods to solve the quadratic equations.
- Factorisation
- Completing the square
- Quadratic formula
We shall discuss the following two methods in detail.
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.
Example:
Find the roots of the quadratic equation \(2x^2 + 4 = 9x\).
Solution:
The given equation is \(2x^2 + 4 = 9x\).
Let us first write the given equation in standard form.
\(2x^2 - 9x + 4 = 0\)
Now, split the middle term by the above procedure.
\(2x^2 - x - 8x + 4 = 0\)
Group the expression into two linear factors by taking the common term outside.
\(x(2x - 1) - 4(2x - 1) = 0\)
\((2x - 1)(x - 4) = 0\)
Now, solve for \(x\) by equating each linear factor to zero.
\(2x - 1 = 0\) or \(x - 4 = 0\)
\(x = \frac{1}{2}\) or \(x = 4\)
Therefore, the roots of \(2x^2 + 4 = 9x\) are \(\frac{1}{2}\) and \(4\).
Solution:
The given equation is \(2x^2 + 4 = 9x\).
Let us first write the given equation in standard form.
\(2x^2 - 9x + 4 = 0\)
Now, split the middle term by the above procedure.
\(2x^2 - x - 8x + 4 = 0\)
Group the expression into two linear factors by taking the common term outside.
\(x(2x - 1) - 4(2x - 1) = 0\)
\((2x - 1)(x - 4) = 0\)
Now, solve for \(x\) by equating each linear factor to zero.
\(2x - 1 = 0\) or \(x - 4 = 0\)
\(x = \frac{1}{2}\) or \(x = 4\)
Therefore, the roots of \(2x^2 + 4 = 9x\) are \(\frac{1}{2}\) and \(4\).
Quadratic formula:
The formula for finding the roots of the quadratic equation \(ax^2 + bx + c = 0\) is:
This formula is known as the quadratic formula.
Example:
Find the roots of \(2x^2 + 3x - 77 = 0\) by using quadratic formula.
Solution:
The given equation is \(2x^2 + 3x - 77 = 0\).
Here, \(a = 2\), \(b = 3\) and \(c = -77\).
Quadratic formula:
Solution:
The given equation is \(2x^2 + 3x - 77 = 0\).
Here, \(a = 2\), \(b = 3\) and \(c = -77\).
Quadratic formula:
Substitute the given values in the formula.
or
or
\(x =\) or \(x = -7\)
Therefore, the roots of the given equation are \(-7\) and .