Solve. [tex]b^2=54-3 b[/tex]
Answer (2)
We solved the equation b 2 = 54 − 3 b by rewriting it in standard quadratic form, factoring it, and finding the roots. The solutions are b = − 9 and b = 6 . Hence, the final answer is b = − 9 , 6 .
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Rewrite the given equation in the standard quadratic form: b 2 + 3 b − 54 = 0 .
Factor the quadratic expression: ( b + 9 ) ( b − 6 ) = 0 .
Set each factor to zero and solve for b : b + 9 = 0 or b − 6 = 0 .
The solutions are b = − 9 and b = 6 , so the final answer is 6 , − 9 .
Explanation
Rewrite the Equation We are given the equation b 2 = 54 − 3 b . Our goal is to solve for b . This looks like a quadratic equation, so let's rearrange it into the standard form a x 2 + b x + c = 0 .
Standard Quadratic Form Add 3 b to both sides of the equation and subtract 54 from both sides to get: b 2 + 3 b − 54 = 0 Now we have a quadratic equation in standard form, where a = 1 , b = 3 , and c = − 54 .
Factoring the Quadratic We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to − 54 and add to 3 . These numbers are 9 and − 6 , since 9 × − 6 = − 54 and 9 + ( − 6 ) = 3 . Therefore, we can factor the quadratic equation as follows: ( b + 9 ) ( b − 6 ) = 0
Solving for b To find the solutions for b , we set each factor equal to zero: b + 9 = 0 or b − 6 = 0 Solving for b in each case, we get: b = − 9 or b = 6 Thus, the solutions are b = − 9 and b = 6 .
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For example, if you are launching a rocket, you can use a quadratic equation to model its height as a function of time and determine when it will reach its maximum height or hit the ground. Understanding how to solve quadratic equations allows us to make predictions and solve problems in these types of scenarios.
