4x ^ 2 – 5x – 12 = 0
Quadratic equations, or equations with a variable raised to the second power, can often be tricky to solve.
Solving Quadratic Equations: The Formula Method
However, with the formula method, you can easily find the roots or solutions of any quadratic equation. In this blog post, we will explore the steps involved in solving a quadratic equation using the formula method and provide some examples for better understanding.
The Formula
The formula for solving a quadratic equation is x = (-b ± √(b^2-4ac)) / 2a where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0.
This formula is also known as the quadratic formula or second-degree formula. It is derived from the quadratic formula theorem which states that any quadratic equation can be solved using this formula.
Step-by-Step Method
To solve a quadratic equation using the formula method, follow these simple steps:
Step 1: Write down the coefficients a, b, and c from the standard form of the quadratic equation.
Step 2: Substitute the values of a, b, and c in the quadratic formula.
Step 3: Simplify the formula using basic arithmetic operations like addition, subtraction, multiplication, and division.
Step 4: Solve for the values of x using the ± symbol to represent both the positive and negative solutions.
Step 5: Check your solutions by substituting the values of x in the original equation. If the equation holds true, then the solutions are correct. Otherwise, recheck your calculations.
Example
Let’s solve the quadratic equation 4x^2 – 5x – 12 = 0 using the formula method.
Step 1: a = 4, b = -5, c = -12
Step 2: x = (-(-5) ± √((-5)^2-4(4)(-12))) / 2(4)
Step 3: x = (5 ± √(25+192)) / 8
Step 4: x = (5 ± √217) / 8
Step 5: x = (5 + √217) / 8 or x = (5 – √217) / 8
Therefore, the solutions of the quadratic equation 4x^2 – 5x – 12 = 0 are x = 3.5 and x = -0.857.
Conclusion
The formula method is a quick and efficient way to solve any quadratic equation. It is especially useful when the equation cannot be easily factored. However, it is always recommended to check your solutions to ensure their accuracy. With enough practice, you can become proficient in solving quadratic equations using the formula method.
