Which shows the correct substitution of the values $a, b$, and $c$ from the equation $0=-3 x^2-2 x+6$ into the quadratic formula?
Quadratic formula: $x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$
A. $x=\frac{-(-2) \pm \sqrt{(-2)^2-4(-3)(6)}}{2(-3)}$
B. $x=\frac{-2 \pm \sqrt{2^2-4(-3)(6)}}{2(-3)}$
C. $x=\frac{-(-2) \pm \sqrt{(-2)^2-4(3)(6)}}{2(3)}$
D. $x=\frac{-2 \pm \sqrt{2^2-4(3)(6)}}{2(3)}$
Answer (2)
Identify the coefficients: a = − 3 , b = − 2 , and c = 6 .
Substitute the values into the quadratic formula: x = 2 ( − 3 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( − 3 ) ( 6 ) .
Compare the result with the given options.
The correct substitution is: x = 2 ( − 3 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( − 3 ) ( 6 ) .
Explanation
Problem Analysis We are given the quadratic equation 0 = − 3 x 2 − 2 x + 6 and the quadratic formula x = 2 a − b ± b 2 − 4 a c . Our goal is to correctly substitute the coefficients a , b , c from the quadratic equation into the quadratic formula.
Identifying Coefficients From the given quadratic equation 0 = − 3 x 2 − 2 x + 6 , we can identify the coefficients as follows:
a = − 3 b = − 2 c = 6
Substituting into the Quadratic Formula Now, we substitute these values into the quadratic formula:
x = 2 a − b ± b 2 − 4 a c = 2 ( − 3 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( − 3 ) ( 6 )
Comparing with the Options Let's compare our result with the given options:
The correct substitution is x = 2 ( − 3 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( − 3 ) ( 6 ) .
Final Answer Therefore, the correct substitution of the values a , b , and c into the quadratic formula is:
x = 2 ( − 3 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( − 3 ) ( 6 )
Examples
The quadratic formula is a powerful tool used in various fields, such as physics and engineering, 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. By correctly substituting the coefficients into the quadratic formula, we can find the time at which the projectile reaches a certain height or the maximum height it attains. This has practical applications in designing ballistics or optimizing the launch angle of a projectile.
The correct substitution of the values a , b , and c into the quadratic formula is represented by option A: x = 2 ( − 3 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( − 3 ) ( 6 ) . This option accurately includes the coefficients identified from the equation. Therefore, option A is the correct choice.
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