(2x%2B1)%3Dx(x%2B5)-is-a-quadratic-equation.jpeg "(x-3)(2x+1)=x(x+5) Is A Quadratic Equation")
Solving Quadratic Equations: (x-3)(2x+1) = x(x+5)
This article will guide you through the process of solving the quadratic equation (x-3)(2x+1) = x(x+5).
Understanding Quadratic Equations
A quadratic equation is a polynomial equation with the highest power of the variable being 2. It can be written in the standard form:
ax² + bx + c = 0
Where 'a', 'b', and 'c' are coefficients, and 'a' cannot be 0.
Solving the Equation
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Expand both sides of the equation:
(x-3)(2x+1) = x(x+5) 2x² - 5x - 3 = x² + 5x
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Simplify by moving all terms to one side:
2x² - 5x - 3 - x² - 5x = 0
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Combine like terms:
x² - 10x - 3 = 0
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Now the equation is in standard quadratic form:
ax² + bx + c = 0
Where a = 1, b = -10, and c = -3
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You can now solve for x using various methods, including:
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Factoring: Try to factor the quadratic expression.
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Quadratic Formula: The formula solves for x directly:
x = (-b ± √(b² - 4ac)) / 2a
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Completing the Square: A method to rewrite the equation in a form where it can be easily solved.
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Finding the Solutions
Applying the quadratic formula:
x = (10 ± √((-10)² - 4 * 1 * -3)) / (2 * 1) x = (10 ± √(112)) / 2 x = (10 ± 4√7) / 2
Therefore, the solutions are:
- x = 5 + 2√7
- x = 5 - 2√7
Conclusion
The equation (x-3)(2x+1) = x(x+5) is indeed a quadratic equation. By expanding, simplifying, and applying appropriate methods like the quadratic formula, we have found the two solutions: x = 5 + 2√7 and x = 5 - 2√7.