How to Make Perfect Square of Quadratic Equation
In mathematics, a quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to zero. Solving quadratic equations is an essential skill in algebra and calculus. One common method to solve quadratic equations is by making them a perfect square. This process, known as completing the square, allows us to easily find the roots of the equation. In this article, we will discuss how to make a perfect square of a quadratic equation.
Understanding the Concept
Before we dive into the steps, it’s crucial to understand the concept of a perfect square. A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be written as 2^2. In the context of quadratic equations, a perfect square refers to the expression (x + h)^2, where h is a constant.
Steps to Make a Perfect Square
To make a perfect square of a quadratic equation, follow these steps:
1. Ensure that the coefficient of the x^2 term is 1. If not, divide the entire equation by the coefficient of the x^2 term.
2. Move the constant term to the right side of the equation by subtracting it from both sides.
3. Divide the coefficient of the x term by 2 and square the result. Add this value to both sides of the equation.
4. Simplify the equation by expanding the perfect square trinomial on the left side.
5. Factor the resulting expression as a product of two binomials.
6. Solve for x by setting each binomial equal to zero and simplifying.
Example
Consider the quadratic equation 2x^2 + 8x – 6 = 0. To make it a perfect square, follow these steps:
1. Divide the entire equation by 2: x^2 + 4x – 3 = 0.
2. Move the constant term to the right side: x^2 + 4x = 3.
3. Divide the coefficient of the x term (4) by 2, which is 2. Square the result (2^2 = 4) and add it to both sides: x^2 + 4x + 4 = 3 + 4.
4. Simplify the equation: (x + 2)^2 = 7.
5. Factor the expression: (x + 2)^2 – 7 = 0.
6. Solve for x: x + 2 = ±√7. Subtract 2 from both sides: x = -2 ± √7.
In conclusion, making a perfect square of a quadratic equation is a useful technique for solving quadratic equations. By following the steps outlined in this article, you can easily transform a quadratic equation into a perfect square and find its roots.
