A quadratic equation is a polynomial equation of degree 2, typically written in the form: $$ ax^2+bx+c=0. $$ where:
Now, we will be focusing on finding the solution of this quadratic equation. Since the quadratic equation has degree 2, it always has 2 solutions that satisfy the equation. The solutions, the values of x, that satisfies the equation are also called the roots of the equation.
Graphical solution of the quadratic equation $ ax^2+bx+c=0 $ is the solution of the two function $ y=ax^2+bx+c $ which represents a parabola and $ y=0 $ which is the equation of the x-axis.
The solutions of the quadratic equation are the x-coordinates where the parabola intersects the x-axis, i.e., the roots of the equation which are given by the following formula, popularly know as quadratic formula.
The nature of the roots is defined by the discriminant function $ b^2 - 4ac $ which is characterized as follows:
The beauty of the graphical solution of the quadratic equation is the actual visual display of these characteristics that we are going to show in a minute.
Various methods exist to solve this quadratic equation, some of them we shall discuss here in detail are listed below.
Square root method:
If $ b = 0 $ in the genereal quadratic equation $ a x^2 + b x + c = 0 $, it takes the form
a $ ax2 + c = 0 $, in such case we use square root method to solve this equation, tha is,
$$ ax ^2 = c $$
$$ x ^2 = \frac{c}{a} $$
$$ x = \pm \sqrt {\frac{c}{a}} $$
Completing square method:
Let's discuss the steps of solving the quadratic equation
\[ ax^2 + bx + c = 0 \]
using the completing the square method:
Step 1:
Divide both sides by 'a' (assuming \( a \neq 0 \)) to make the coefficient of \( x^2 \) equal to 1:
\[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \]
Step 2:
Move the constant term \( \frac{c}{a} \) to the other side:
\[ x^2 + \frac{b}{a}x = -\frac{c}{a} \]
Step 3:
Complete the square for the quadratic term. To do this, we need to add and subtract \( \left(\frac{b}{2a}\right)^2 \):
\[ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} \]
Step 4: Recognize the complete square term:
\[ \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} \]
Step 5:
To isolate \( x \), we need to get rid of the constant term on the left side. To do this, we add \( \left(\frac{b}{2a}\right)^2 \) to both sides:
\[ \left(x + \frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 -\frac{c}{a} \]
Step 6:
Now, take the square root on both sides:
\[ x + \frac{b}{2a} = \pm \sqrt{\left(\frac{b}{2a}\right)^2 -\frac{c}{a}} \]
Step 7:
Subtract \( \frac{b}{2a} \) from both sides and simplify the expression under the radical sign:
\[ x = -\frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} \]
Step 9:
Since \( 4a^2 \) is in the denominator of the square root, it can be taken out as \( 2a \):
\[ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \]
Step 10:
Combine the fractions:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This is the quadratic formula obtained using the completing the square method.
The roots are:
First:
\[ x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \]
Second:
\[ x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]
Formula method:
General Quadratic Equation: a x2 + b x + c = 0.
Graphical method:
Graphical solution of the quadratic equation
$$ ax^2 + bx +c =0 $$
is the solution of the two function
$$ y = ax^2 + bx +c $$
which represents a parabola and
$$ y = 0 $$
which is the equation of the x-axis.
The intersection of these two functions is the two points (if the solution is real) on the x-axis, is the solution of the quadratic equation:
$$ ax^2 + bx +c =0 $$
The solution is given by,
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The nature of the roots is defined by the discriminant function $ b^2 - 4ac $ which is characterized as follows: