$x^2=3x+18$
Rewrite the equation in factored form.
$\square$ = 0
(Factor completely.)
The solution set is {$\square$}.
(Use a comma to separate answers as needed. Type each solution only once.)
Answer (2)
Rewrite the equation in standard form: x 2 − 3 x − 18 = 0 .
Factor the quadratic expression: ( x − 6 ) ( x + 3 ) = 0 .
Set each factor to zero and solve for x : x = 6 or x = − 3 .
The solution set is { − 3 , 6 } .
Explanation
Rewrite the Equation Let's solve the quadratic equation x 2 = 3 x + 18 . Our first goal is to rewrite the equation in the standard form of a quadratic equation, which is a x 2 + b x + c = 0 . This will allow us to factor the quadratic expression and find the solutions for x .
Standard Form To rewrite the equation in standard form, we subtract 3 x and 18 from both sides of the equation:
x 2 − 3 x − 18 = 3 x + 18 − 3 x − 18
This simplifies to:
x 2 − 3 x − 18 = 0
Factor the Quadratic Now, we need to factor the quadratic expression x 2 − 3 x − 18 . We are looking for two numbers that multiply to − 18 and add to − 3 . These numbers are − 6 and 3 . Therefore, we can factor the quadratic expression as follows:
( x − 6 ) ( x + 3 ) = 0
Solve for x To find the solutions for x , we set each factor equal to zero and solve for x :
x − 6 = 0 or x + 3 = 0
Solving these equations gives us:
x = 6 or x = − 3
Solution Set Therefore, the solution set is { − 3 , 6 } .
Examples
Quadratic equations are incredibly useful in various real-life scenarios. For instance, they can help calculate the trajectory of a ball, determine the optimal dimensions for a garden to maximize area, or even model financial investments. Imagine you're designing a rectangular garden and want its area to be 18 square meters with a specific relationship between the length and width. By setting up a quadratic equation, you can find the exact dimensions needed to achieve your desired area. This blend of math and practical application highlights the power of quadratic equations in problem-solving.
To solve the equation x 2 = 3 x + 18 , we first rewrite it in standard form as x 2 − 3 x − 18 = 0 . After factoring, we find that the solutions are x = 6 and x = − 3 , resulting in the solution set { − 3 , 6 }.
;
