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Solving the Equation (2x-1)(x+3) = 4
This article will guide you through the steps of solving the equation (2x-1)(x+3) = 4.
1. Expand the Equation
First, we need to expand the left side of the equation by multiplying the two binomials:
(2x-1)(x+3) = 2x² + 6x - x - 3
Simplifying, we get:
2x² + 5x - 3 = 4
2. Move all terms to one side
To solve this quadratic equation, we need to set it equal to zero. This is done by subtracting 4 from both sides:
2x² + 5x - 7 = 0
3. Factor the Quadratic Equation
Now we need to factor the quadratic expression on the left side. This can be done by finding two numbers that add up to 5 (the coefficient of the x term) and multiply to -14 (the product of the coefficient of the x² term and the constant term).
The numbers 7 and -2 satisfy these conditions:
7 + (-2) = 5 7 * (-2) = -14
Therefore, we can rewrite the equation as:
(2x - 2)(x + 7) = 0
4. Solve for x
For the product of two terms to be zero, at least one of the terms must be zero. So we have two possible solutions:
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2x - 2 = 0 Solving for x, we get: x = 1
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x + 7 = 0 Solving for x, we get: x = -7
5. Verification
To verify our solutions, we can substitute them back into the original equation:
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For x = 1: (2(1) - 1)(1 + 3) = (1)(4) = 4 This solution is correct.
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For x = -7: (2(-7) - 1)(-7 + 3) = (-15)(-4) = 60 ≠ 4 This solution is incorrect.
Conclusion
Therefore, the only valid solution to the equation (2x-1)(x+3) = 4 is x = 1.