What is the solution of the equation [tex]x^2-14 x+67=0[/tex]?
Answer (2)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 1 , b = − 14 , and c = 67 .
Simplify the expression to x = 2 14 ± 196 − 268 = 2 14 ± − 72 .
Rewrite using the imaginary unit: x = 2 14 ± i 72 = 2 14 ± 6 i 2 .
Divide by 2 to get the final answer: x = 7 ± 3 i 2 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 14 x + 67 = 0 . Our goal is to find the solution(s) for x . We can use the quadratic formula to solve for x .
Applying the Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c for a quadratic equation of the form a x 2 + b x + c = 0 . In this case, a = 1 , b = − 14 , and c = 67 .
Substitution Substitute these values into the quadratic formula: x = 2 ( 1 ) − ( − 14 ) ± ( − 14 ) 2 − 4 ( 1 ) ( 67 ) .
Simplifying Simplify the expression: x = 2 14 ± 196 − 268 .
More Simplifying Further simplify: x = 2 14 ± − 72 .
Introducing the Imaginary Unit Rewrite the square root of the negative number using the imaginary unit i : x = 2 14 ± i 72 .
Simplifying the Radical Simplify the radical: 72 = 36 ⋅ 2 = 6 2 .
Back-Substitution Substitute this back into the equation: x = 2 14 ± 6 i 2 .
Final Solution Divide both terms in the numerator by 2: x = 7 ± 3 i 2 . Therefore, the solution to the equation x 2 − 14 x + 67 = 0 is x = 7 ± 3 i 2 .
Complex Conjugates The solutions are complex conjugates: x 1 = 7 + 3 i 2 and x 2 = 7 − 3 i 2 .
Examples
Quadratic equations appear in various fields, such as physics (projectile motion), engineering (designing structures), and economics (modeling costs and profits). For example, when designing a bridge, engineers use quadratic equations to calculate the curve of an arch or the trajectory of a cable. By solving these equations, they can ensure the structure's stability and safety. Similarly, in economics, quadratic functions can model the relationship between the price of a product and the quantity demanded, helping businesses optimize their pricing strategies.
The solutions to the equation x 2 − 14 x + 67 = 0 are complex numbers, specifically x = 7 ± 3 i 2 . These represent two complex solutions: x 1 = 7 + 3 i 2 and x 2 = 7 − 3 i 2 .
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