Quadratic equation:

A quadratic equation is a polynomial equation of degree 2, typically written in the form: $$ ax^2+bx+c=0. $$ where:

Examples:

  1.   $ x^2−5x+6=0 $

  2.  $ 2x^2+3x−2=0 $

Solution of Quadratic Equation:

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.

Visualization of quadratic equation:

The quadratic function $ y=ax^2+bx+c $ is represented graphically as a parabola.

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.

$$ x = \frac {- b \pm \sqrt{b^2 - 4 a c}}{2a} $$

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.

  1. If $ b^2 - 4ac > 0 $;
    The graph of the quadratic function intersect the x-axis at 2 points shown in the following figure displays 2 distinct roots. Image failed
  2. If $ b^2 - 4ac = 0 $;
    The graph of the quadratic function intersect the x-axis at 1 point shown in the following figure, displays only one but repeated roots. Image failed
  3. If $ b^2 - 4ac < 0 $
    The graph of the quadratic function does not intersect the x-axis shown in the following figure, displays no real roots so the roots are complex. Image failed
  4. Methods of solving quadratic equations:

    Various methods exist to solve this quadratic equation, some of them we shall discuss here in detail are listed below.

    1. 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, $$ ax ^2 = c $$ $$ x ^2 = \frac{c}{a} $$ $$ x = \pm \sqrt {\frac{c}{a}} $$

    2. Factor method:
      The quadratic equation: $$ ax^2 + bx +c =0 $$ can be factorized into $$ (x + m)(x + n) = 0 $$ Considering 2 real numbers m and n, Such that, $$ m + n = b; $$ and $$ m*n = a*c $$
    3. 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: \[ x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]

    4. Formula method:
      General Quadratic Equation: a x2 + b x + c = 0.

      Quadratic formula:
      $$ x = \frac {- b \pm \sqrt{b^2 - 4 a c}}{2a} $$

    5. 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:

      • If the discriminant is positive, that is, $ b^2 - 4ac > 0 $ there are two distinct real roots;
      • If the discriminant is zero, that is, $ b^2 - 4ac = 0 $ there is one real root (a repeated root);
      • If the discriminant is negative, that ia, $ b^2 - 4ac< 0 $ there are no real roots (two complex roots).

    Practical applicatios:

    1. Physics:
    2. Engineering:
    3. Finance:
    4. ...