Which are the solutions of the quadratic equation?
[tex]x^2=9 x+6[/tex]
A. [tex]\frac{-9-\sqrt{105}}{2}, \frac{-9+\sqrt{105}}{2}[/tex]
B. [tex]\frac{-9-\sqrt{57}}{2}, \frac{-9+\sqrt{57}}{2}[/tex]
C. [tex]\frac{9-\sqrt{105}}{2}, \frac{9+\sqrt{105}}{2}[/tex]
D. [tex]\frac{9-\sqrt{57}}{2}, \frac{9+\sqrt{57}}{2}[/tex]
Answer (1)
Rewrite the quadratic equation in the standard form: x 2 − 9 x − 6 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values a = 1 , b = − 9 , and c = − 6 into the formula and simplify.
The solutions are x = 2 9 − 105 and x = 2 9 + 105 , so the answer is 2 9 − 105 , 2 9 + 105 .
Explanation
Problem Analysis We are given a quadratic equation x 2 = 9 x + 6 and asked to find its solutions from a list of options.
Rewrite the Equation First, we rewrite the equation in the standard form a x 2 + b x + c = 0 . This gives us x 2 − 9 x − 6 = 0 . Here, a = 1 , b = − 9 , and c = − 6 .
Apply Quadratic Formula Next, we use the quadratic formula to find the solutions for x . The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c
Substitute Values Substituting the values of a , b , and c into the formula, we get: x = 2 ( 1 ) − ( − 9 ) ± ( − 9 ) 2 − 4 ( 1 ) ( − 6 )
Simplify Simplifying the expression, we have: x = 2 9 ± 81 + 24 = 2 9 ± 105
Solutions Thus, the two solutions are x 1 = 2 9 − 105 and x 2 = 2 9 + 105 .
Final Answer Comparing our solutions with the given options, we find that the correct answer is 2 9 − 105 , 2 9 + 105 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, 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 100 square meters and you know that the length must be 5 meters longer than the width, you can set up a quadratic equation to find the dimensions of the garden.
