Using the quadratic formula to solve $4 x^2-3 x+9=2 x+1$, what are the values of $x$?
$
\frac{9 \pm \sqrt{158}}{18}$
$
\frac{5 \pm \sqrt{153}}{8}$
$
\frac{5 \pm \sqrt{103 i}}{8}$
$
\frac{1 \pm \sqrt{153}}{6}$
Answer (2)
Rewrite the 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 = 8 5 ± − 103 .
Simplify to find the values of x : x = 8 5 ± 103 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 − 3 x + 9 = 2 x + 1 . Subtracting 2 x and 1 from both sides, we get 4 x 2 − 3 x − 2 x + 9 − 1 = 0 , which simplifies to 4 x 2 − 5 x + 8 = 0 .
Identify coefficients Now, we identify the coefficients a , b , and c in the quadratic equation 4 x 2 − 5 x + 8 = 0 . We have a = 4 , b = − 5 , and c = 8 .
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 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( 4 ) ( 8 )
x = 8 5 ± 25 − 128
x = 8 5 ± − 103
x = 8 5 ± 103 i
Final Answer Therefore, the values of x are 8 5 + 103 i and 8 5 − 103 i .
Examples
Quadratic equations are used in various fields, such as physics to calculate the trajectory of a projectile, engineering to design structures, and economics to model supply and demand curves. For example, if you are launching a rocket, you can use a quadratic equation to model its height as a function of time, taking into account gravity and initial velocity. By solving the quadratic equation, you can determine when the rocket will reach its maximum height or when it will hit the ground. This allows engineers to optimize the launch parameters for the desired outcome.
To solve the quadratic equation 4 x 2 − 3 x + 9 = 2 x + 1 , we transform it into standard form 4 x 2 − 5 x + 8 = 0 and apply the quadratic formula. This results in two complex solutions: x = 8 5 ± 103 i . The chosen multiple choice option is 8 5 ± 103 i .
;
