Which shows the correct substitution of the values $a, b$, and $c$ from the equation $-2=-x+x^2-4$ into the quadratic formula?
Quadratic formula: $x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$
A. $x=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-4)}}{2(1)}$
B. $x=\frac{-1 \pm \sqrt{1^2-4(-1)(-4)}}{2(-1)}$
C. $x=\frac{-1 \pm \sqrt{(1)^2-4(-1)(-2)}}{2(-1)}$
D. $x=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-2)}}{2(1)}$
Answer (2)
Rewrite the given equation in the standard quadratic form: x 2 − x − 2 = 0 .
Identify the coefficients: a = 1 , b = − 1 , and c = − 2 .
Substitute the values into the quadratic formula: x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 2 ) .
The correct substitution is x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 2 ) .
Explanation
Rewrite the Equation We are given the equation − 2 = − x + x 2 − 4 and the quadratic formula x = 2 a − b ± b 2 − 4 a c . Our goal is to find the correct substitution of the values a , b , and c from the given equation into the quadratic formula. First, we need to rewrite the given equation in the standard quadratic form a x 2 + b x + c = 0 .
Standard Quadratic Form To rewrite the equation − 2 = − x + x 2 − 4 in the standard form, we add 2 to both sides to get 0 = − x + x 2 − 4 + 2 , which simplifies to 0 = x 2 − x − 2 . Thus, we have x 2 − x − 2 = 0 .
Identify Coefficients Now we can identify the coefficients a , b , and c . Comparing x 2 − x − 2 = 0 with the standard form a x 2 + b x + c = 0 , we have a = 1 , b = − 1 , and c = − 2 .
Substitute into Quadratic Formula Next, we substitute these values into the quadratic formula x = 2 a − b ± b 2 − 4 a c . Substituting a = 1 , b = − 1 , and c = − 2 , we get x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 2 ) .
Find the Correct Option Comparing our result x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 2 ) with the given options, we find that the correct substitution is x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 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 designs. For example, when designing a bridge, engineers use quadratic equations to model the curve of the suspension cables, ensuring stability and minimizing stress. By correctly substituting the parameters into the quadratic formula, they can determine critical points and make informed decisions about the bridge's structure.
The correct values of a, b, and c from the equation − 2 = − x + x 2 − 4 are a = 1 , b = − 1 , and c = − 2 . Substituting these into the quadratic formula gives us the expression x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 2 ) . Thus, the correct option is D .
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