Final Answer:
Solving the equation 2y(y+5)=48 yields two solutions: y = 4 and y = -12.
Step-by-step explanation:
To solve the equation 2y(y+5)=48, we first expand the equation: 2y² + 10y = 48. Rearranging terms brings the equation to the standard quadratic form: 2y² + 10y - 48 = 0. To solve for y, we can either factorize or apply the quadratic formula. Factoring the equation gives us (2y - 8)(y + 6) = 0. Setting each factor equal to zero yields y = 4 and y = -12 as the solutions. These values satisfy the equation, making both y = 4 and y = -12 valid solutions.
Utilizing the distributive property to expand the equation 2y(y+5)=48 helps simplify the expression to a quadratic equation, 2y² + 10y = 48. Rearranging terms and setting the equation to zero by subtracting 48 from both sides leads to the quadratic equation 2y² + 10y - 48 = 0. We can then factorize this equation by finding two numbers that multiply to give -96 (the product of the coefficients of y² and the constant term) and add up to 10 (the coefficient of the y term). The factors are (2y - 8)(y + 6) = 0, yielding the solutions y = 4 and y = -12.
Therefore, after factoring the quadratic equation derived from the initial expression, the solutions for y are found to be y = 4 and y = -12, both satisfying the equation 2y(y+5)=48 when substituted back into the original equation. These solutions represent the values of y that make the equation true and can be verified by substituting them back into the original equation to confirm their validity.