To determine whether expression \( a \) is equivalent to expression \( b \) and to transform \( a \) to \( b \), we need to simplify and manipulate the expression step by step. However, the expressions need to be clarified a bit as you have presented them. Let's assume the expressions are:
- Expression \( a \): \( x + 4x - 2 \)
- Expression \( b \): \( x^2 + 2x - 8 \)
Steps to Transform Expression \( a \) into Expression \( b \)
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Combine Like Terms in Expression \( a \): \[ a = x + 4x - 2 = 5x - 2 \]
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Set Up an Equation to Check for Equivalence: We want to check if \( 5x - 2 \) is equivalent to \( x^2 + 2x - 8 \). We'll set the expressions equal to each other: \[ 5x - 2 = x^2 + 2x - 8 \]
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Rearrange the Equation: Move all terms to one side of the equation: \[ 0 = x^2 + 2x - 8 - 5x + 2 \] Which simplifies to: \[ 0 = x^2 - 3x - 6 \]
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Formulate as a Standard Quadratic Equation: Rearranging the above, we have: \[ x^2 - 3x - 6 = 0 \]
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Check for Roots (Factoring or Using Quadratic Formula): This equation can be factored or solved using the quadratic formula:
- The discriminant \( D = b^2 - 4ac \) for \( a=1, b=-3, c=-6 \): \[ D = (-3)^2 - 4(1)(-6) = 9 + 24 = 33 \] Since the discriminant is positive, there are two real roots.
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Find the Roots Using Quadratic Formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{33}}{2} \] These roots show that \( a \) and \( b \) are not equivalent for all \( x \).
Conclusion
After analyzing both expressions, we find that \( a = 5x - 2 \) is not equivalent to \( b = x^2 + 2x - 8 \), as they produce different outputs given the same input values of \( x \).
If the expressions \( a \) and \( b \) you were referring to are different, please clarify, and I can assist you in verifying their equivalence!