Which two values of [tex]$x$[/tex] are roots of the polynomial below?
[tex]$x^2+3 x-5$[/tex]
A. [tex]$x=\frac{-3-\sqrt{11}}{2}$[/tex]
B. [tex]$x=\frac{3-\sqrt{-11}}{2}$[/tex]
C. [tex]$x=\frac{-3+\sqrt{11}}{2}$[/tex]
D. [tex]$x=\frac{-3+\sqrt{29}}{2}$[/tex]
E. [tex]$x=\frac{-3-\sqrt{29}}{2}$[/tex]
F. [tex]$x=\frac{3+\sqrt{-11}}{2}$[/tex]
Answer (1)
Use the quadratic formula x = 2 a − b ± b 2 − 4 a c to find the roots of the quadratic equation.
Substitute a = 1 , b = 3 , and c = − 5 into the formula.
Simplify the expression to find the two roots: x = 2 − 3 + 29 and x = 2 − 3 − 29 .
The two values of x that are roots of the polynomial are x = 2 − 3 + 29 and x = 2 − 3 − 29 .
Explanation
Understanding the Problem We are given the quadratic polynomial x 2 + 3 x − 5 and asked to find its roots from the given options.
Applying the Quadratic Formula To find the roots of the quadratic polynomial, we can use the quadratic formula. The quadratic formula states that for a quadratic equation of the form a x 2 + b x + c = 0 , the roots are given by: x = 2 a − b ± b 2 − 4 a c In our case, a = 1 , b = 3 , and c = − 5 .
Calculating the Roots Substituting the values of a , b , and c into the quadratic formula, we get: x = 2 ( 1 ) − 3 ± 3 2 − 4 ( 1 ) ( − 5 ) x = 2 − 3 ± 9 + 20 x = 2 − 3 ± 29
Identifying the Correct Options So the two roots are: x = 2 − 3 + 29 and x = 2 − 3 − 29
Final Answer Comparing these roots with the given options, we find that they match options D and E.
Examples
Understanding quadratic equations and their roots is crucial in various fields, such as physics and engineering. For instance, when analyzing the trajectory of a projectile, the roots of a quadratic equation can determine the points where the projectile hits the ground. Similarly, in electrical engineering, quadratic equations are used to analyze circuits and determine the values of components that satisfy certain conditions. By mastering the quadratic formula and its applications, students can gain a deeper understanding of these real-world phenomena and develop problem-solving skills that are valuable in many disciplines.
