Using the quadratic formula to solve [tex]4 x^2-3 x+9=2 x+1[/tex], what are the values of [tex]x[/tex]?
A. [tex]\frac{1 \pm \sqrt{159} i}{8}[/tex]
B. [tex]\frac{5 \pm \sqrt{153} i}{8}[/tex]
C. [tex]\frac{5 \pm \sqrt{103} i}{8}[/tex]
D. [tex]\frac{1 \pm \sqrt{153}}{8}[/tex]
Answer (2)
Rewrite the given quadratic equation in the standard form: 4 x 2 − 5 x + 8 = 0 .
Identify the coefficients: a = 4 , b = − 5 , c = 8 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 2 ( 4 ) 5 ± ( − 5 ) 2 − 4 ( 4 ) ( 8 ) = 8 5 ± − 103 .
Simplify to obtain the complex solutions: x = 8 5 ± 103 i .
Explanation
Understanding the Problem We are given the quadratic equation 4 x 2 − 3 x + 9 = 2 x + 1 . Our goal is to solve for x using the quadratic formula. The quadratic formula is a powerful tool that provides the solutions to any quadratic equation in the form a x 2 + b x + c = 0 .
Rewriting the Equation First, we need to rewrite the given equation in the standard form a x 2 + b x + c = 0 . Subtract 2 x and 1 from both sides of the equation: 4 x 2 − 3 x + 9 − 2 x − 1 = 0 4 x 2 − 5 x + 8 = 0
Identifying Coefficients Now, we can identify the coefficients a , b , and c :
a = 4 , b = − 5 , and c = 8 .
Applying the Quadratic Formula Next, we apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c Substitute the values of a , b , and c into the formula: x = 2 ( 4 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( 4 ) ( 8 ) x = 8 5 ± 25 − 128 x = 8 5 ± − 103
Simplifying the Expression Since the discriminant (the value inside the square root) is negative, we have complex solutions. We can rewrite the square root of a negative number using the imaginary unit i , where i = − 1 :
x = 8 5 ± 103 i
Final Answer Therefore, the values of x are 8 5 + 103 i and 8 5 − 103 i . We can write this as: x = 8 5 ± 103 i
Conclusion Comparing our result with the given options, we see that the correct answer is 8 5 ± 103 i .
Examples
The quadratic formula is not just an abstract concept; it has real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as a ball thrown in the air or a rocket launched into space. By knowing the initial velocity and angle of launch, they can predict the range and maximum height of the projectile. Similarly, in finance, quadratic equations can be used to model investment growth and calculate the time it takes for an investment to reach a certain value, considering compound interest and other factors. These applications demonstrate the practical importance of understanding and solving quadratic equations.
The solutions to the quadratic equation 4 x 2 − 3 x + 9 = 2 x + 1 are x = 8 5 ± 103 i . Therefore, the correct answer is option B. The complex solutions arise because the discriminant is negative.
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