Which equation shows the quadratic formula used correctly to solve $5 x^2+3 x-4=0$ for $x ?$
A. $x=\frac{-3 \pm \sqrt{(3)^2-4(5)(-4)}}{2(5)}$
B. $x=\frac{3 \pm \sqrt{(3)^2+4(5)(-4)}}{2(5)}$
C. $x=\frac{3 \pm \sqrt{(3)^2-4(5)(-4)}}{2(5)}$
D. $x=\frac{-3 \pm \sqrt{(3)^2+4(5)(-4)}}{2(5)}$
Answer (1)
Identify the coefficients: a = 5 , b = 3 , and c = − 4 .
Substitute the coefficients into the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
The correct equation is: x = 2 ( 5 ) − 3 ± ( 3 ) 2 − 4 ( 5 ) ( − 4 ) .
Therefore, the solution is x = 2 ( 5 ) − 3 ± ( 3 ) 2 − 4 ( 5 ) ( − 4 ) .
Explanation
Understanding the Problem We are given the quadratic equation 5 x 2 + 3 x − 4 = 0 and asked to identify the correct application of the quadratic formula to solve for x . The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 .
Applying the Quadratic Formula In our equation, 5 x 2 + 3 x − 4 = 0 , we can identify the coefficients as follows: a = 5 , b = 3 , and c = − 4 . Now, we substitute these values into the quadratic formula: x = 2 ( 5 ) − 3 ± ( 3 ) 2 − 4 ( 5 ) ( − 4 )
Simplifying the Equation Simplifying the expression under the square root: ( 3 ) 2 − 4 ( 5 ) ( − 4 ) = 9 + 80 = 89 So, the equation becomes: x = 10 − 3 ± 89
Identifying the Correct Option Comparing this with the given options, we see that the correct equation is: x = 2 ( 5 ) − 3 ± ( 3 ) 2 − 4 ( 5 ) ( − 4 )
Examples
The quadratic formula is a fundamental tool in algebra, used to solve equations that model various real-world scenarios. For instance, imagine you're designing a suspension bridge and need to calculate the precise curve of the supporting cables. This curve can often be described by a quadratic equation, and using the quadratic formula, engineers can determine the exact points where the cables need to be anchored to ensure the bridge's stability and safety. This ensures that the bridge can withstand the forces acting upon it, preventing potential collapses.
