Solve the equation using the quadratic formula. [tex]4 x^2-x=-5[/tex] The solution set is { }. (Simplify your answer. Type an exact answer, using radicals and [tex]i[/tex] as needed. expression. Use a comma to separate answers as needed.)
Answer (2)
Rewrite the equation in standard form: 4 x 2 − x + 5 = 0 .
Identify coefficients: a = 4 , b = − 1 , c = 5 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 8 1 ± − 79 .
Simplify to find the solution set: { 8 1 + 8 79 i , 8 1 − 8 79 i } .
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 4 x 2 − x = − 5 . Adding 5 to both sides, we get 4 x 2 − x + 5 = 0 .
Identify the coefficients Now, we can identify the coefficients a , b , and c . In the equation 4 x 2 − x + 5 = 0 , we have a = 4 , b = − 1 , and c = 5 .
Apply the quadratic formula Next, we apply the quadratic formula, which is given by x = 2 a − b ± b 2 − 4 a c . Substituting the values of a , b , and c , we get:
x = 2 ( 4 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 4 ) ( 5 )
Simplify the expression Now, we simplify the expression:
x = 8 1 ± 1 − 80
x = 8 1 ± − 79
Since the discriminant is negative, we have complex solutions. We can rewrite the square root of -79 as − 79 = 79 i . Therefore,
x = 8 1 ± 79 i
Express the solution set Thus, the two solutions are x = 8 1 + 79 i and x = 8 1 − 79 i . We can write the solution set as { 8 1 + 8 79 i , 8 1 − 8 79 i } .
Final Answer Therefore, the solution set is { 8 1 + 8 79 i , 8 1 − 8 79 i } .
Examples
Quadratic equations are incredibly useful in physics, engineering, and even economics. For example, if you're designing a bridge, you need to calculate the parabolic arc that distributes weight evenly. Imagine you're launching a rocket; the trajectory often follows a quadratic path. By solving quadratic equations, engineers can determine the rocket's maximum height and range. Similarly, economists use quadratic functions to model cost and revenue curves, helping businesses find the optimal production level to maximize profit. The quadratic formula is a versatile tool that helps solve these real-world problems by finding the roots of quadratic equations, which represent key points in these scenarios.
The equation 4 x 2 − x = − 5 can be rewritten as 4 x 2 − x + 5 = 0 with coefficients a = 4 , b = − 1 , and c = 5 . Applying the quadratic formula, the solutions are x = 8 1 + 8 79 i and x = 8 1 − 8 79 i , which form the solution set { \frac{1}{8} + \frac{\sqrt{79}}{8}i, \frac{1}{8} - \frac{\sqrt{79}}{8}i }.
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