Solve.
[tex]w(w-6)=16[/tex]
Answer (2)
To solve the equation w ( w − 6 ) = 16 , we expand it to w 2 − 6 w − 16 = 0 and factor it to find the solutions, which are w = 8 and w = − 2 .
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Expand the equation: w ( w − 6 ) = w 2 − 6 w = 16 .
Rearrange to standard quadratic form: w 2 − 6 w − 16 = 0 .
Factor the quadratic: ( w − 8 ) ( w + 2 ) = 0 .
Solve for w : w = 8 or w = − 2 . The solutions are 8 , − 2 .
Explanation
Problem Analysis We are given the equation w ( w − 6 ) = 16 and asked to solve for w . This is a quadratic equation, which we can solve by rearranging it into standard form, factoring, and then finding the roots.
Expanding the Equation First, we expand the left side of the equation: w ( w − 6 ) = w 2 − 6 w So the equation becomes: w 2 − 6 w = 16
Setting the Equation to Zero Next, we subtract 16 from both sides to set the equation to zero: w 2 − 6 w − 16 = 0
Factoring the Quadratic Now, we factor the quadratic expression. We are looking for two numbers that multiply to -16 and add to -6. These numbers are -8 and 2. So we can factor the quadratic as: ( w − 8 ) ( w + 2 ) = 0
Finding the Solutions To find the solutions for w , we set each factor equal to zero: w − 8 = 0 or w + 2 = 0
The Solutions Solving these equations, we get: w = 8 or w = − 2
Final Answer Therefore, the solutions to the equation w ( w − 6 ) = 16 are w = 8 and w = − 2 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden to maximize area, or modeling the growth of a population. For example, if you are designing a rectangular garden with a fixed perimeter and want to find the dimensions that give the maximum area, you might end up solving a quadratic equation. Similarly, in physics, the height of a projectile over time can be modeled by a quadratic equation, allowing you to predict when it will hit the ground.
