Given the equation $3 x^2-7 x-1=0$, find $\alpha-\beta$.
Answer (1)
Use Vieta's formulas to find the sum and product of the roots: α + β = 3 7 and α β = − 3 1 .
Apply the identity ( α − β ) 2 = ( α + β ) 2 − 4 α β .
Substitute the values to get ( α − β ) 2 = 9 61 .
Take the square root to find α − β = 3 61 .
Explanation
Understanding the Problem We are given the quadratic equation 3 x 2 − 7 x − 1 = 0 . Our goal is to find the difference between the roots, denoted as α − β .
Recalling Vieta's Formulas Recall Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation of the form a x 2 + b x + c = 0 , the sum of the roots is given by α + β = − a b and the product of the roots is given by α β = a c .
Applying Vieta's Formulas to Our Equation In our equation, 3 x 2 − 7 x − 1 = 0 , we have a = 3 , b = − 7 , and c = − 1 . Applying Vieta's formulas, we find that the sum of the roots is α + β = − 3 − 7 = 3 7 and the product of the roots is α β = 3 − 1 = − 3 1 .
Using the Identity to Find the Difference To find α − β , we can use the identity ( α − β ) 2 = ( α + β ) 2 − 4 α β . Substituting the values we found earlier, we get
( α − β ) 2 = ( 3 7 ) 2 − 4 ( − 3 1 ) = 9 49 + 3 4 = 9 49 + 9 12 = 9 61 .
Finding the Square Root Taking the square root of both sides, we have α − β = ± 9 61 = ± 3 61 . Therefore, the difference between the roots is either 3 61 or − 3 61 . Since the problem does not specify which root is larger, we can provide the positive value as the answer.
Final Answer Thus, α − β = 3 61 .
Examples
Understanding the roots of a quadratic equation and their relationships is crucial in various fields, such as physics and engineering. For instance, when analyzing projectile motion, the roots of a quadratic equation can represent the times at which the projectile reaches a certain height. The difference between these roots can then provide insights into the duration of the projectile's flight. By understanding Vieta's formulas and the relationships between roots, engineers and physicists can solve real-world problems related to motion, optimization, and system analysis.
