Solve. [tex]$4 x^2-x-5=0$[/tex]
Answer (2)
The solutions to the quadratic equation 4 x 2 − x − 5 = 0 can be found using the quadratic formula, yielding results of x = 1.25 and x = − 1 . The calculation involves identifying coefficients and substituting them into the formula to solve for x . Hence, the final answers are x = 1.25 and x = − 1 .
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Identify the coefficients: a = 4 , b = − 1 , c = − 5 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values and simplify: x = 8 1 ± 81 .
Calculate the two solutions: x 1 = 1.25 and x 2 = − 1 . The solutions to the equation are x = 1.25 , − 1 .
Explanation
Understanding the Problem We are given the quadratic equation 4 x 2 − x − 5 = 0 . Our goal is to find the values of x that satisfy this equation. We can use the quadratic formula to solve for x .
Identifying Coefficients and the Quadratic Formula 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 . In our case, a = 4 , b = − 1 , and c = − 5 .
Substituting Values into the Formula Now, we substitute the values of a , b , and c into the quadratic formula: x = 2 ( 4 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 4 ) ( − 5 ) x = 8 1 ± 1 + 80 x = 8 1 ± 81 x = 8 1 ± 9
Calculating the Solutions We have two possible solutions for x :
x 1 = 8 1 + 9 = 8 10 = 4 5 = 1.25 x 2 = 8 1 − 9 = 8 − 8 = − 1
Final Answer Therefore, the solutions to the quadratic equation 4 x 2 − x − 5 = 0 are x = 1.25 and x = − 1 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 20 square meters and you know that one side is 3 meters longer than the other, you can use a quadratic equation to find the dimensions of the garden.
