(x-5).jpeg "(x-2)(x-5)")
Factoring and Solving (x-2)(x-5) = 0
The expression (x-2)(x-5) represents a factored quadratic equation. Let's break down how to understand and solve it.
Understanding the Factored Form
- Factored form: This form highlights the roots or solutions of the equation. Each factor represents a linear expression that equals zero when the entire equation equals zero.
- Individual factors: (x-2) and (x-5) are the two factors.
- Zero Product Property: The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Solving for x
To find the solutions for x, we use the Zero Product Property:
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Set each factor equal to zero:
- x - 2 = 0
- x - 5 = 0
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Solve for x in each equation:
- x = 2
- x = 5
The Solutions
Therefore, the solutions to the equation (x-2)(x-5) = 0 are x = 2 and x = 5.
Expanding the Equation
We can expand the factored form to get the standard quadratic equation:
(x-2)(x-5) = x² - 7x + 10
This confirms that the solutions we found for the factored form are also the solutions to the expanded quadratic equation.
Significance of the Solutions
The solutions x = 2 and x = 5 represent the x-intercepts of the parabola defined by the quadratic equation y = x² - 7x + 10. These are the points where the parabola crosses the x-axis.