Which equation has the solutions [tex]$x=\frac{5 \pm 2 \sqrt{7}}{3}$[/tex]?
[tex]$3 x^2-5 x+7=0$[/tex]
[tex]$3 x^2-5 x-1=0$[/tex]
[tex]$3 x^2-10 x+6=0$[/tex]
[tex]$3 x^2-10 x-1=0$[/tex]
Answer (1)
Calculate the sum of the roots: x 1 + x 2 = 3 10 .
Calculate the product of the roots: x 1 x 2 = − 3 1 .
Form the quadratic equation: x 2 − 3 10 x − 3 1 = 0 .
Multiply by 3 to get the final equation: 3 x 2 − 10 x − 1 = 0 .
Explanation
Understanding the Problem We are given the solutions x = 3 5 ± 2 7 and we need to find the quadratic equation that has these solutions.
Setting up the Quadratic Equation Let the roots of the quadratic equation be x 1 = 3 5 + 2 7 and x 2 = 3 5 − 2 7 . A quadratic equation with roots x 1 and x 2 can be written in the form ( x − x 1 ) ( x − x 2 ) = 0 . Expanding this, we get x 2 − ( x 1 + x 2 ) x + x 1 x 2 = 0 .
Calculating Sum and Product of Roots We need to find the sum and product of the roots. The sum of the roots is: x 1 + x 2 = 3 5 + 2 7 + 3 5 − 2 7 = 3 5 + 2 7 + 5 − 2 7 = 3 10 The product of the roots is: x 1 x 2 = 3 5 + 2 7 ⋅ 3 5 − 2 7 = 9 ( 5 + 2 7 ) ( 5 − 2 7 ) = 9 5 2 − ( 2 7 ) 2 = 9 25 − 4 ( 7 ) = 9 25 − 28 = 9 − 3 = − 3 1
Forming the Quadratic Equation Substituting the sum and product of the roots into the quadratic equation, we get: x 2 − 3 10 x − 3 1 = 0 To eliminate the fractions, we multiply the entire equation by 3: 3 x 2 − 10 x − 1 = 0
Finding the Correct Option Comparing the resulting equation with the given options, we find that the correct equation is 3 x 2 − 10 x − 1 = 0 .
Examples
Understanding quadratic equations is crucial in many fields, such as physics and engineering. For example, 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 bridge's stability. Similarly, in projectile motion, quadratic equations are used to calculate the trajectory of a projectile, helping to determine its range and maximum height.
