Solve using the quadratic formula:
$2 x^2-x-7=-2$
Answer (1)
Rewrite the equation in standard form: 2 x 2 − x − 5 = 0 .
Identify coefficients: a = 2 , b = − 1 , c = − 5 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c = 4 1 ± 41 .
The solution is x = 4 1 ± 4 41 .
Explanation
Rewrite the equation First, we need to rewrite the given equation in the standard quadratic form, which is a x 2 + b x + c = 0 . The given equation is 2 x 2 − x − 7 = − 2 . Adding 2 to both sides, we get: 2 x 2 − x − 5 = 0
Identify the coefficients Now, we identify the coefficients a , b , and c in the quadratic equation 2 x 2 − x − 5 = 0 . We have: a = 2 b = − 1 c = − 5
Apply the quadratic formula Next, we apply the quadratic formula, which is given by: x = f r a c − b p m s q r t b 2 − 4 a c 2 a Substituting the values of a , b , and c , we get: x = f r a c − ( − 1 ) p m s q r t ( − 1 ) 2 − 4 ( 2 ) ( − 5 ) 2 ( 2 ) x = f r a c 1 p m s q r t 1 + 40 4 x = f r a c 1 p m s q r t 41 4
Simplify the expression So the solutions are: x = f r a c 1 4 + f r a c s q r t 41 4 q u a d and q u a d x = f r a c 1 4 − f r a c s q r t 41 4 x = f r a c 1 p m s q r t 41 4
Compare with proposed solutions Comparing the calculated solutions with the proposed solutions, we see that the correct solution is: x = f r a c 1 4 p m f r a c s q r t 41 4 The other options are incorrect.
Examples
The quadratic formula is a powerful tool used in various fields, such as physics, engineering, and economics, to solve problems involving quadratic equations. For example, in physics, it can be used to determine the trajectory of a projectile, where the height of the projectile is described by a quadratic equation. In engineering, it can be used to design structures or circuits that satisfy certain quadratic relationships. In economics, it can be used to model cost and revenue functions to find the break-even point.
