Use factoring to solve the quadratic equation. Check by substitution or by using a graphing calculator. [tex]5 x^2=33 x+14[/tex] Rewrite the equation in factored form. $\square =0$ (Factor completely.) The solution set is {$\square$}. (Use commas to separate answers as needed. Type each solution only once.)
Answer (2)
Rewrite the equation in standard form: 5 x 2 − 33 x − 14 = 0 .
Factor the quadratic expression: ( 5 x + 2 ) ( x − 7 ) = 0 .
Set each factor to zero and solve for x : 5 x + 2 = 0 or x − 7 = 0 .
The solutions are x = − 5 2 and x = 7 , so the solution set is { − 5 2 , 7 } .
Explanation
Problem Analysis We are given the quadratic equation 5 x 2 = 33 x + 14 . Our goal is to solve this equation by factoring. This involves rewriting the equation in the standard quadratic form, factoring the quadratic expression, and then finding the values of x that make the equation true.
Rewrite in Standard Form First, we need to rewrite the equation in the standard form a x 2 + b x + c = 0 . Subtract 33 x and 14 from both sides of the equation: 5 x 2 − 33 x − 14 = 0
Find Factors Now, we need to factor the quadratic expression 5 x 2 − 33 x − 14 . We are looking for two numbers that multiply to 5 × ( − 14 ) = − 70 and add up to − 33 . These numbers are − 35 and 2 . So, we can rewrite the middle term using these numbers: 5 x 2 − 35 x + 2 x − 14 = 0
Factor by Grouping Next, we factor by grouping: 5 x ( x − 7 ) + 2 ( x − 7 ) = 0
Factor out Common Term Now, we factor out the common term ( x − 7 ) :
( 5 x + 2 ) ( x − 7 ) = 0
Set Factors to Zero To find the solutions, we set each factor equal to zero and solve for x :
5 x + 2 = 0 or x − 7 = 0
Solve First Equation Solving 5 x + 2 = 0 for x :
5 x = − 2 x = − 5 2
Solve Second Equation Solving x − 7 = 0 for x :
x = 7
Final Solution Therefore, the solution set is { − 5 2 , 7 } .
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. In business, quadratic equations can be used to analyze profit margins, calculate optimal pricing strategies, and predict revenue based on sales volume. Understanding how to solve these equations allows us to make informed decisions and predictions in both scientific and everyday contexts.
We rewrote the quadratic equation 5 x 2 = 33 x + 14 in standard form as 5 x 2 − 33 x − 14 = 0 and factored it to obtain ( 5 x + 2 ) ( x − 7 ) = 0 . The solutions to the equation are x = − 5 2 and x = 7 , giving us the solution set of { -\frac{2}{5}, 7 }.
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