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 (1)
Rewrite the 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 First, we need to rewrite the given equation − 2 = − x + x 2 − 4 in the standard quadratic form a x 2 + b x + c = 0 . To do this, we add 2 to both sides of the equation to get 0 = − x + x 2 − 4 + 2 , which simplifies to 0 = x 2 − x − 2 .
Identify coefficients Now, we can identify the coefficients a , b , and c in the quadratic equation a x 2 + b x + c = 0 . In our case, 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 with the given options, we see 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 arch's shape and ensure its stability. By correctly substituting the parameters into the quadratic formula, they can determine critical points and make informed decisions about the bridge's structure, ensuring it can withstand various loads and environmental conditions. This ensures safety and efficiency in the design process.
