Which two values of $x$ are roots of the polynomial below?
$4 x^2-6 x+1$
A. $x=\frac{6+\sqrt{20}}{8}$
B. $x=\frac{-6-\sqrt{52}}{16}$
c. $x=\frac{-8-\sqrt{28}}{6}$
D. $x=\frac{-8+\sqrt{28}}{6}$
E. $x=\frac{6-\sqrt{20}}{8}$
F. $x=\frac{6+\sqrt{52}}{16}$
Answer (1)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 4 , b = − 6 , and c = 1 .
Calculate the discriminant: b 2 − 4 a c = ( − 6 ) 2 − 4 ( 4 ) ( 1 ) = 36 − 16 = 20 .
Find the roots: x = 8 6 ± 20 .
The two roots are x = 8 6 + 20 and x = 8 6 − 20 , corresponding to options A and E. Therefore, the answer is x = 8 6 + 20 , x = 8 6 − 20 .
Explanation
Understanding the Problem We are given a quadratic polynomial 4 x 2 − 6 x + 1 and six possible roots. Our goal is to identify which two of the given values are the roots of the polynomial.
Applying the Quadratic Formula We can use the quadratic formula to find the roots of the polynomial. The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 4 , b = − 6 , and c = 1 .
Calculating the Roots Substituting the values of a , b , and c into the quadratic formula, we get: x = 2 ( 4 ) − ( − 6 ) ± ( − 6 ) 2 − 4 ( 4 ) ( 1 ) x = 8 6 ± 36 − 16 x = 8 6 ± 20
Identifying the Correct Options Now, we simplify the roots: x = 8 6 + 20 and x = 8 6 − 20 Comparing these roots with the given options, we can see that they match options A and E.
Final Answer Therefore, the two values of x that are roots of the polynomial 4 x 2 − 6 x + 1 are: x = 8 6 + 20 and x = 8 6 − 20 These correspond to options A and E.
Examples
Understanding quadratic equations and their roots is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. The roots of the equation help determine the points where the arch meets the supports, ensuring the structural integrity of the bridge. Similarly, in physics, projectile motion can be described using quadratic equations, where the roots indicate the launch and landing points of the projectile.
