Rewrite the equation in factored form.
[tex]5 x^2=44 x+9[/tex]
[tex]\square = 0[/tex]
(Factor completely.)
The solution set is {[tex]\square[/tex]}.
(Use commas to separate answers as needed. Type each solution only once.)
Answer (2)
Rewrite the equation in standard form: 5 x 2 − 44 x − 9 = 0 .
Factor the quadratic expression: ( 5 x + 1 ) ( x − 9 ) = 0 .
Set each factor to zero and solve for x : x = − 5 1 or x = 9 .
Write the solution set: { − 5 1 , 9 } .
Explanation
Problem Analysis We are given the quadratic equation 5 x 2 = 44 x + 9 . Our goal is to rewrite this equation in factored form and then find the solution set.
Rewrite in Standard Form First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . To do this, we subtract 44 x and 9 from both sides of the equation: 5 x 2 − 44 x − 9 = 0
Factor the Quadratic Expression Now, we need to factor the quadratic expression 5 x 2 − 44 x − 9 . We are looking for two binomials of the form ( A x + B ) ( C x + D ) such that their product equals 5 x 2 − 44 x − 9 .
We can use the factoring method. We need to find two numbers that multiply to 5 × − 9 = − 45 and add up to − 44 . These numbers are − 45 and 1 .
So, we can rewrite the middle term as − 44 x = − 45 x + x . Then, we have: 5 x 2 − 45 x + x − 9 = 0 Now, we can factor by grouping: 5 x ( x − 9 ) + 1 ( x − 9 ) = 0 ( 5 x + 1 ) ( x − 9 ) = 0
Write the Factored Form Now that we have the factored form of the equation, we can write the equation in the required format: ( 5 x + 1 ) ( x − 9 ) = 0
Solve for x To find the solution set, we set each factor equal to zero and solve for x :
5 x + 1 = 0 ⇒ 5 x = − 1 ⇒ x = − 5 1
x − 9 = 0 ⇒ x = 9
So, the solutions are x = − 5 1 and x = 9 .
Write the Solution Set Finally, we write the solution set as a set containing these two values: { − 5 1 , 9 }
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to determine its maximum height and range. Engineers use quadratic equations to design bridges and arches, ensuring structural integrity and stability. Financial analysts also employ quadratic equations to model investment growth and predict market trends, making informed decisions about asset allocation and risk management.
To factor the equation 5 x 2 = 44 x + 9 , we first rewrite it in standard form as 5 x 2 − 44 x − 9 = 0 . Factoring gives us ( 5 x + 1 ) ( x − 9 ) = 0 , which results in the solution set \ i g { − 5 1 , 9 } .
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