Solve the equation that is in quadratic form.
$(x-2)^2-11(x-2)+19=-5$
Answer (1)
Substitute y = x − 2 to simplify the equation.
Solve the quadratic equation y 2 − 11 y + 24 = 0 by factoring to get y = 3 or y = 8 .
Substitute back x − 2 for y and solve for x .
The solutions are x = 5 and x = 10 , so the final answer is 5 , 10 .
Explanation
Understanding the Problem We are given the equation ( x − 2 ) 2 − 11 ( x − 2 ) + 19 = − 5 . This equation is in quadratic form, which means we can solve it by making a substitution to simplify it.
Making a Substitution Let's substitute y = x − 2 . Then the equation becomes y 2 − 11 y + 19 = − 5 .
Rearranging the Equation Now, we rearrange the equation to get a standard quadratic equation: y 2 − 11 y + 19 + 5 = 0 , which simplifies to y 2 − 11 y + 24 = 0 .
Factoring the Quadratic We can factor this quadratic equation as ( y − 3 ) ( y − 8 ) = 0 .
Solving for y Solving for y , we get y = 3 or y = 8 .
Substituting Back Now we substitute back x − 2 for y . So we have x − 2 = 3 or x − 2 = 8 .
Solving for x Solving for x in each case, we get x = 3 + 2 = 5 or x = 8 + 2 = 10 . Therefore, the solutions are x = 5 and x = 10 .
Verifying the Solutions To verify the solutions, we can plug them back into the original equation. For x = 5 , we have ( 5 − 2 ) 2 − 11 ( 5 − 2 ) + 19 = ( 3 ) 2 − 11 ( 3 ) + 19 = 9 − 33 + 19 = − 5 , which is correct. For x = 10 , we have ( 10 − 2 ) 2 − 11 ( 10 − 2 ) + 19 = ( 8 ) 2 − 11 ( 8 ) + 19 = 64 − 88 + 19 = − 5 , which is also correct.
Final Answer The solutions to the equation ( x − 2 ) 2 − 11 ( x − 2 ) + 19 = − 5 are x = 5 and x = 10 .
Examples
Imagine you are designing a bridge and need to calculate the stress on a particular support beam. The stress can be modeled by a quadratic equation. By solving this equation, you can determine the critical load values that the beam can withstand before it fails. Understanding how to solve quadratic equations is crucial in engineering to ensure the safety and stability of structures.
