Use the Quadratic Formula to solve the given equation on the interval [ $0 , \frac { \pi } { 2 }$ );then use a graphing utility to approximate the angle x.Round answers to three decimal places. $75 \cos ^ { 2 } x - 34 \cos x + 3 = 0$
A)x = 1.229,1.433
B)x = 1.227,1.422
C)x = 1.233,1.469
D)x = 1.231,1.451
E)x = 1.235,1.480

Question

Use the Quadratic Formula to solve the given equation on the interval [ $0 , \frac { \pi } { 2 }$ );then use a graphing utility to approximate the angle x.Round answers to three decimal places.​ $75 \cos ^ { 2 } x - 34 \cos x + 3 = 0$ ​
A)x = 1.229,1.433
B)​x = 1.227,1.422
C)​x = 1.233,1.469
D)​x = 1.231,1.451
E)​x = 1.235,1.480

Quizplus · Accepted Answer

To solve the given equation $75 \cos ^ { 2 } x - 34 \cos x + 3 = 0$ using the Quadratic Formula, let $y = \cos x$. The equation becomes $75y^2 - 34y + 3 = 0$. Applying the Quadratic Formula, $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 75$, $b = -34$, and $c = 3$, we find the values of $y$ (which are the values of $\cos x$). Then, we use a graphing utility or inverse cosine function to find the values of $x$ within the given interval $[0, \frac{\pi}{2})$. The correct values, rounded to three decimal places, match the option D.

Use the Quadratic Formula to Solve the Given Equation on the Interval