What is the solution of the equation [tex]$x^2+8 x=4$[/tex]?
A. [tex]$x=-4 \pm 2 \sqrt{3}$[/tex]
B. [tex]$x=-4 \pm 2 \sqrt{5}$[/tex]
C. [tex]$x=-4 \pm 5 \sqrt{2}$[/tex]
D. [tex]$x=-4 \pm 4 \sqrt{5}$[/tex]
Answer (1)
Rewrite the equation in standard quadratic form: x 2 + 8 x − 4 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute a = 1 , b = 8 , and c = − 4 into the formula and simplify.
The solutions are x = − 4 ± 2 5 .
Explanation
Problem Analysis We are given the quadratic equation x 2 + 8 x = 4 . Our goal is to find the solution(s) for x . To do this, we will rewrite the equation in standard quadratic form and then use the quadratic formula.
Rewrite in Standard Form First, we rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . Subtracting 4 from both sides of the equation x 2 + 8 x = 4 , we get x 2 + 8 x − 4 = 0 . Here, a = 1 , b = 8 , and c = − 4 .
Apply Quadratic Formula Next, we apply the quadratic formula x = 2 a − b ± b 2 − 4 a c to find the solutions for x . Plugging in the values a = 1 , b = 8 , and c = − 4 , we have
x = 2 ( 1 ) − 8 ± 8 2 − 4 ( 1 ) ( − 4 ) x = 2 − 8 ± 64 + 16 x = 2 − 8 ± 80
Simplify the Expression Now, we simplify the expression. We can simplify 80 as follows:
80 = 16 × 5 = 16 × 5 = 4 5
So, we have
x = 2 − 8 ± 4 5
Dividing both terms in the numerator by 2, we get
x = − 4 ± 2 5
Final Answer Therefore, the solutions to the equation x 2 + 8 x = 4 are x = − 4 + 2 5 and x = − 4 − 2 5 .
Examples
Quadratic equations are not just abstract math; they appear in various real-world scenarios. For example, imagine you're designing a rectangular garden. You know you want the length to be 8 meters more than the width, and the total area to be 4 square meters. By setting up a quadratic equation, you can determine the exact dimensions (width and length) of the garden to meet your specifications. This involves solving for the unknown variable, which represents the width of the garden.
