Solve the equation using the quadratic formula.

[tex]5 x^2=2 x+4[/tex]

The solution set is { }. (Simplify your answer. Type an exact answer, using radical answers as needed.)

Rewrite the equation in standard form: 5 x 2 − 2 x − 4 = 0 .
Identify coefficients: a = 5 , b = − 2 , c = − 4 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ = 10 2 ± 84 ​ ​ = 5 1 ± 21 ​ ​ .
The solution set is { 5 1 − 21 ​ ​ , 5 1 + 21 ​ ​ } ​ .

Explanation

Rewrite the equation First, we need to rewrite the given equation in the standard quadratic form, which is a x 2 + b x + c = 0 . The given equation is 5 x 2 = 2 x + 4 . Subtracting 2 x and 4 from both sides, we get 5 x 2 − 2 x − 4 = 0 .

Identify the coefficients Now, we identify the coefficients a , b , and c . In the equation 5 x 2 − 2 x − 4 = 0 , we have a = 5 , b = − 2 , and c = − 4 .

Apply the quadratic formula Next, we apply the quadratic formula, which is given by x = f r a c − b p m s q r t b 2 − 4 a c 2 a . Substituting the values of a , b , and c , we have:


x = 2 ( 5 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 5 ) ( − 4 ) ​ ​
x = 10 2 ± 4 + 80 ​ ​
x = 10 2 ± 84 ​ ​
x = 10 2 ± 4 × 21 ​ ​
x = 10 2 ± 2 21 ​ ​
x = 5 1 ± 21 ​ ​

State the solution set Therefore, the two solutions are x = 5 1 + 21 ​ ​ and x = 5 1 − 21 ​ ​ . The solution set is { 5 1 − 21 ​ ​ , 5 1 + 21 ​ ​ } .

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 them to predict investment returns and manage risk. Understanding quadratic equations empowers you to solve problems related to optimization, motion, and growth in numerous fields.

Answered by GinnyAnswer | 2025-07-03

To solve the equation 5 x 2 = 2 x + 4 , we rewrite it in standard form as 5 x 2 − 2 x − 4 = 0 . By applying the quadratic formula, we find the solutions to be 5 1 − 21 ​ ​ and 5 1 + 21 ​ ​ . The solution set is { \frac{1 - \sqrt{21}}{5}, \frac{1 + \sqrt{21}}{5} }.
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Answered by Anonymous | 2025-07-04