Solve the equation using the quadratic formula.
[tex]2 x(x-2)=-6 x+4[/tex]
The solution set is $\square$ , (Simplify your answer, including any radicals and [tex]i[/tex] as needed. Use comma to separate answers as needed.)
Answer (1)
Rewrite the given equation 2 x ( x − 2 ) = − 6 x + 4 in the standard quadratic form: x 2 + x − 2 = 0 .
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 1 , b = 1 , and c = − 2 .
Simplify the expression to find the two solutions: x = 2 − 1 ± 9 = 2 − 1 ± 3 .
The solution set is 1 , − 2 .
Explanation
Rewrite the Equation We are given the equation 2 x ( x − 2 ) = − 6 x + 4 and asked to solve it using the quadratic formula. Our first step is to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 .
Expand the Left Side First, expand the left side of the equation: 2 x 2 − 4 x = − 6 x + 4
Move Terms to One Side Next, move all terms to the left side of the equation: 2 x 2 − 4 x + 6 x − 4 = 0
Combine Like Terms Combine like terms: 2 x 2 + 2 x − 4 = 0
Simplify the Equation Divide the entire equation by 2 to simplify: x 2 + x − 2 = 0
Identify Coefficients Now we can identify the coefficients: a = 1 , b = 1 , and c = − 2 . We will use these values in the quadratic formula.
State the Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c
Substitute Values Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 1 ) − 1 ± 1 2 − 4 ( 1 ) ( − 2 )
Simplify the Square Root Simplify the expression under the square root: 1 2 − 4 ( 1 ) ( − 2 ) = 1 + 8 = 9
Calculate the Square Root Calculate the square root: 9 = 3
Substitute Back Substitute the square root back into the quadratic formula: x = 2 − 1 ± 3
Find the Solutions Find the two solutions: x 1 = 2 − 1 + 3 = 2 2 = 1 x 2 = 2 − 1 − 3 = 2 − 4 = − 2
State the Solution Set Therefore, the solution set is {1, -2}.
Examples
The quadratic formula is a powerful tool used in various fields, such as physics and engineering, to solve problems involving parabolic trajectories or optimizing system performance. For example, when designing a bridge, engineers use quadratic equations to model the curve of the suspension cables, ensuring structural integrity and stability. Similarly, in physics, the trajectory of a projectile, like a ball thrown in the air, can be described by a quadratic equation, allowing us to predict its range and maximum height.
