Method of Solving a Quadratic Equation - Quadratic Formula (Shreedharacharya's Rule)

For the quadratic equation ax² + bx + c = 0, a ≠ 0; the expression b² − 4ac is called the discriminant and is, in general, denoted by the letter 'D'.

Thus, discriminant D = b² − 4ac.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

For \[ y = ax^{2} + bx + c, \] the real roots are the x-intercepts of the parabola.

Solve: \[ m^{2} - 14m + 13 = 0. \]

Step 1: Compare with the standard form to find coefficients

Given equation: \[ m^{2} - 14m + 13 = 0. \]

Comparing with \[ am^{2} + bm + c = 0, \] we get:

\[ a = 1, \quad b = -14, \quad c = 13 \]

Step 2: Calculate the discriminant ( \[ b^{2} - 4ac \] )

\[ b^{2} - 4ac = (-14)^{2} - 4(1)(13) \]

\[ = 196 - 52 = 144 \]

Step 3: Apply the quadratic formula

The formula for roots is:

\[ m = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

Substituting the values:

\[ m = \frac{-(-14) \pm \sqrt{144}}{2(1)} = \frac{14 \pm 12}{2} \]

Question: Solve the quadratic equation \[ x^{2} - 2x - 3 = 0 \] graphically.

Solution:

Step 1: Rearrange the equation

Rewrite the given equation into an equivalent form: \[ x^{2} = 2x + 3. \]

Step 2: Formulate two separate equations for graphing

Split the equation into two functions: \[ y = x^{2} \] and \[ y = 2x + 3. \]

Step 3: Plot and find intersection points

Draw the graphs for both functions using coordinate tables.

Let \[ y = x^2 = 2x + 3. \] Let us draw graph of \[ y = x^2 \] and \[ y = 2x + 3 \]

\[ y = x^2 \]

\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 3 & 2 & 1 & 0 & -1 & -2 & -3 \\ \hline y & 9 & 4 & 1 & 0 & 1 & 4 & 9 \\ \hline \end{array} \]

\[ y = 2x + 3 \]

\[ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 1 & -2 \\ \hline y & 1 & 3 & 5 & -1 \\ \hline \end{array} \]

Observe that the two graphs intersect each other at the points \[ (-1, 1) \] and \[ (3, 9). \]

Step 4: Extract the roots

The x-coordinates of the intersection points represent the solutions to the equation.

Therefore, the solutions are \[ x = -1 \] or \[ x = 3. \]

1. Write the given equation in the standard form

   ax² + bx + c = 0

2. Identify the values of a, b, and c.

3. Find the value of the discriminant

   D = b² − 4ac

4. Substitute the values of a, b, and D in the formula

   \[x=\frac{-b-\sqrt{b^2-4ac}}{2a}\]​

5. Simplify to obtain the roots.