A quadratic equation in the variable \(x\) is an equation of the form \(ax^2+bx+c =0\), where \(a, b,c\) are real numbers, \(a \neq 0\). The degree of the quadratic equation is \(2\).
Example:
\(x^2 + 9x +15 =0\).
It is of the form \(ax^2+bx+c =0\).
Therefore, the given equation is a quadratic equation.
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 \(2x^2 + 4 = 9x\). 2x 2 +4=9
Solution:
The given equation is \(2x^2 + 4 = 9x\)
.
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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.
\(x(2x -1) -4(2x -1) =0\)
\((2x -1)(x -4) =0\)
\\(2x -1(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 \(4\).
Important!
Any quadratic equation can have at most two roots.
Let the graph of the parabola be \(y = ax^2 + bx + c\). Then, the solutions of the quadratic equation are the \(x\) coordinates of the intersection of the curve with \(X\) axis.
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.