How to Find Roots of a Quadratic Equation

A quadratic equation is a second-degree polynomial typically written in the form:
$ax^2 + bx + c = 0$
where $a$, $b$, and $c$ are coefficients, and $x$ represents the variable.

Quadratic Formula

The most common method to find the roots of a quadratic equation is by using the quadratic formula:
$x = frac{-b pm sqrt{b^2 – 4ac}}{2a}$

Steps to Use the Quadratic Formula

1. Identify Coefficients: Determine the values of $a$, $b$, and $c$ from the equation. For example, if the equation is $2x^2 + 3x – 5 = 0$, then $a = 2$, $b = 3$, and $c = -5$
2. Calculate the Discriminant: Compute the discriminant using the formula
   $b^2 – 4ac$. The discriminant helps determine the nature of the roots.
   • If the discriminant is positive, there are two distinct real roots.
   • If it is zero, there is one real root (also called a repeated root).
   • If it is negative, the roots are complex (not real).
3. Apply the Quadratic Formula: Substitute $a$, $b$, and $c$ into the quadratic formula to find the roots.

Example

Let’s solve the quadratic equation $2x^2 + 3x – 5 = 0$

1. Identify Coefficients: Here, $a = 2$, $b = 3$, and $c = -5$
2. Calculate the Discriminant: $b^2 – 4ac = 3^2 – 4(2)(-5) = 9 + 40 = 49$ (a positive discriminant, so two real roots).
3. Apply the Quadratic Formula:

   $x = frac{-3 pm sqrt{49}}{4} = frac{-3 pm 7}{4}$

   • $x_1 = frac{-3 + 7}{4} = frac{4}{4} = 1$
   • $x_2 = frac{-3 – 7}{4} = frac{-10}{4} = -2.5$

Thus, the roots of $2x^2 + 3x – 5 = 0$ are $x = 1$ and $x = -2.5$

Conclusion

Understanding the quadratic formula and how to apply it is essential for solving quadratic equations. This method is reliable and works for any quadratic equation, providing real or complex roots based on the discriminant.

Citations

1. 1. Khan Academy – Quadratic Equations
2. 2. Math is Fun – Quadratic Equation
3. 3. Purplemath – Solving Quadratic Equations