(x-5)%3D0-standard-form.jpeg "(x+2)(x-5)=0 Standard Form")
Solving Quadratic Equations: From Factored Form to Standard Form
This article will explore how to convert a quadratic equation in factored form to its standard form. We'll use the example of (x + 2)(x - 5) = 0 to illustrate the process.
Understanding Factored Form
The equation (x + 2)(x - 5) = 0 is in factored form. This means the quadratic expression is presented as a product of two linear expressions. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is crucial for solving equations in factored form.
Converting to Standard Form
To convert to standard form, we need to expand the factored expression and simplify.
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Expand the product: (x + 2)(x - 5) = 0
- Using the distributive property (or FOIL method):
- x * (x - 5) + 2 * (x - 5) = 0
- x² - 5x + 2x - 10 = 0
- Using the distributive property (or FOIL method):
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Combine like terms:
- x² - 3x - 10 = 0
Now the equation is in standard form, which is ax² + bx + c = 0, where a, b, and c are constants.
The Standard Form: ax² + bx + c = 0
In our example, a = 1, b = -3, and c = -10.
Solving for x
The standard form allows us to use various methods to solve for x. In this case, we could:
- Factoring: We can factor the quadratic expression to find its roots.
- Quadratic Formula: This formula provides a direct solution for the roots of any quadratic equation.
- Completing the Square: This method allows us to rewrite the equation in a way that enables us to easily find the roots.
Conclusion
Converting a quadratic equation from factored form to standard form is a fundamental step in solving quadratic equations. The standard form makes it easier to apply various solution methods and ultimately find the values of x that satisfy the equation.