Solve for $y$. $2 y^2+23 y+63=(y+7)^2$
Answer (2)
Expand the right side of the equation: ( y + 7 ) 2 = y 2 + 14 y + 49 .
Simplify the equation to y 2 + 9 y + 14 = 0 .
Factor the quadratic expression: ( y + 2 ) ( y + 7 ) = 0 .
Solve for y : y = − 2 or y = − 7 . The solutions are − 2 , − 7 .
Explanation
Problem Analysis We are given the equation 2 y 2 + 23 y + 63 = ( y + 7 ) 2 and asked to solve for y .
Expanding the Right Side First, we expand the right side of the equation: ( y + 7 ) 2 = y 2 + 2 ( 7 ) y + 7 2 = y 2 + 14 y + 49
Rewriting the Equation Now we rewrite the original equation with the expanded right side: 2 y 2 + 23 y + 63 = y 2 + 14 y + 49
Setting the Equation to Zero Next, we subtract the right side from the left side to set the equation to zero: 2 y 2 + 23 y + 63 − ( y 2 + 14 y + 49 ) = 0
Simplifying the Equation Now we simplify the equation by combining like terms: ( 2 y 2 − y 2 ) + ( 23 y − 14 y ) + ( 63 − 49 ) = 0 y 2 + 9 y + 14 = 0
Factoring the Quadratic We factor the quadratic expression: We are looking for two numbers that multiply to 14 and add to 9. These numbers are 2 and 7. ( y + 2 ) ( y + 7 ) = 0
Solving for y Finally, we solve for y by setting each factor to zero: y + 2 = 0 or y + 7 = 0
The Solutions The solutions are: y = − 2 or y = − 7
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when calculating the trajectory of a projectile, you often encounter quadratic equations that describe the object's height as a function of time. By solving these equations, engineers can determine the range and maximum height of the projectile, which is essential for designing accurate targeting systems or optimizing the performance of sports equipment. Similarly, in economics, quadratic equations can model cost and revenue functions, helping businesses find the optimal production level to maximize profit.
To solve the equation 2 y 2 + 23 y + 63 = ( y + 7 ) 2 , we first expand the right side and rearrange it to form a quadratic equation y 2 + 9 y + 14 = 0 . This factors to ( y + 2 ) ( y + 7 ) = 0 , providing solutions of y = − 2 and y = − 7 .
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