Use the quadratic formula to solve the equation.
[tex]$-x^2-5 x+6=0$[/tex]
Answer (1)
Identify the coefficients: a = − 1 , b = − 5 , and c = 6 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = − 2 5 ± 49 .
Calculate the two solutions: x = − 6 and x = 1 . The final answer is − 6 , 1 .
Explanation
Understanding the Problem and the Quadratic Formula We are given the quadratic equation − x 2 − 5 x + 6 = 0 and asked to solve it using the quadratic formula. The quadratic formula is a general method for finding the roots (solutions) of any quadratic equation in the form a x 2 + b x + c = 0 , where a , b , and c are constants. The formula is given by:
x = 2 a − b ± b 2 − 4 a c
In our equation, we can identify the coefficients as a = − 1 , b = − 5 , and c = 6 . We will substitute these values into the quadratic formula to find the solutions for x .
Substituting Values into the Formula Now, we substitute the values of a , b , and c into the quadratic formula:
x = 2 ( − 1 ) − ( − 5 ) ± ( − 5 ) 2 − 4 ( − 1 ) ( 6 )
Let's simplify the expression step by step.
Simplifying the Expression First, we simplify the terms inside the square root:
( − 5 ) 2 − 4 ( − 1 ) ( 6 ) = 25 + 24 = 49
So, the expression becomes:
x = − 2 5 ± 49
Since 49 = 7 , we have:
x = − 2 5 ± 7
Finding the Solutions Now, we find the two possible values for x :
x 1 = − 2 5 + 7 = − 2 12 = − 6
x 2 = − 2 5 − 7 = − 2 − 2 = 1
So, the solutions are x = − 6 and x = 1 .
Final Answer Therefore, the solutions to the quadratic equation − x 2 − 5 x + 6 = 0 are x = − 6 and x = 1 .
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to determine its maximum height and range. They are also used in engineering to design bridges and arches, ensuring structural stability. In finance, quadratic equations can help calculate investment returns and model economic trends. Understanding how to solve quadratic equations provides a foundation for tackling complex problems in physics, engineering, and economics.
