To find the roots of a quadratic equation, which is in the form ax² + bx + c = 0, you can use two primary methods: the quadratic formula and factorization.
Using the Quadratic Formula
The quadratic formula is a universal method that works for any quadratic equation. It states that the roots (solutions for x) are given by:
x = (-b ± √ (b² - 4ac)) / 2a
Where:
- a is the coefficient of the x² term.
- b is the coefficient of the x term.
- c is the constant term.
Steps to apply the quadratic formula:
- Identify a, b, and c: From your equation ax² + bx + c = 0, determine the values of a, b, and c.
- Substitute the values: Plug the identified values of a, b, and c into the quadratic formula.
- Calculate the discriminant: Compute the expression b² - 4ac. This part, known as the discriminant, determines the nature of the roots.
- If b² - 4ac > 0, there are two distinct real roots.
- If b² - 4ac = 0, there is exactly one real root (a repeated root).
- If b² - 4ac < 0, there are two complex roots (not real).
- Calculate the roots: Perform the remaining calculations in the formula to find the two possible values of x.
Example: Consider the equation 2x² - 5x + 2 = 0.
Here, a=2, b=-5, and c=2.
Using the quadratic formula:
x = (5 ± √((-5)² - 4 2 2)) / (2 * 2)
x = (5 ± √(25 - 16)) / 4
x = (5 ± √9) / 4
x = (5 ± 3) / 4
So the roots are x = (5+3)/4 = 2 and x = (5-3)/4 = 1/2
Using Factorization
If the quadratic expression ax² + bx + c can be factored into two linear expressions, then setting each factor to zero will give you the roots. This method is not always applicable, but when it works it is often quicker.
Steps for factorization:
- Factor the quadratic: Rewrite ax² + bx + c as (px + q)(rx + s).
- Set each factor to zero: Set each of the linear factors equal to zero: px+q=0 and rx+s=0
- Solve each equation for x: Solve for x in each of the linear equations to get the roots of the quadratic.
Example: Consider x² - 5x + 6 = 0.
This can be factored as (x-2)(x-3)=0.
Setting each factor to zero gives:
x - 2 = 0 which implies x = 2
x - 3 = 0 which implies x = 3
So the roots are x=2 and x=3.
Summary Table
Method | Applicability | Process |
---|---|---|
Quadratic Formula | Always works | Substitute coefficients into the formula: x = (-b ± √ (b² - 4ac)) / 2a |
Factorization | Only if factorable | Factor the quadratic expression into linear terms and set each term to zero. |
Both methods provide ways to find the roots of any quadratic equation. Choose the one you find more convenient or suitable for the equation at hand.