To determine which values are solutions to the given equation:
[tex]\[ 4x^2 + 20x + 25 = 49 \][/tex]
We need to follow these steps:
1. Simplify the equation to standard form by subtracting 49 from both sides:
[tex]\[ 4x^2 + 20x + 25 - 49 = 0 \][/tex]
[tex]\[ 4x^2 + 20x - 24 = 0 \][/tex]
2. Solve the quadratic equation:
[tex]\[ 4x^2 + 20x - 24 = 0 \][/tex]
To find the solutions, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In our equation, [tex]\( a = 4 \)[/tex], [tex]\( b = 20 \)[/tex], and [tex]\( c = -24 \)[/tex].
Calculating the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 20^2 - 4(4)(-24) \][/tex]
[tex]\[ \Delta = 400 + 384 \][/tex]
[tex]\[ \Delta = 784 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-20 \pm \sqrt{784}}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{-20 \pm 28}{8} \][/tex]
This gives two possible solutions:
[tex]\[ x = \frac{-20 + 28}{8} = \frac{8}{8} = 1 \][/tex]
[tex]\[ x = \frac{-20 - 28}{8} = \frac{-48}{8} = -6 \][/tex]
The solutions to the equation [tex]\( 4x^2 + 20x + 25 = 49 \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -6 \)[/tex].
Now, let's check the given options:
A. [tex]\( x = -1 \)[/tex]
- Substitute [tex]\( x = -1 \)[/tex]:
[tex]\[ 4(-1)^2 + 20(-1) + 25 = 4 - 20 + 25 = 9 \][/tex]
This does not satisfy the equation [tex]\( 4x^2 + 20x + 25 = 49 \)[/tex].
- Not a solution.
B. [tex]\( x = 6 \)[/tex]
- Substitute [tex]\( x = 6 \)[/tex]:
[tex]\[ 4(6)^2 + 20(6) + 25 = 144 + 120 + 25 = 289 \][/tex]
This does not satisfy the equation [tex]\( 4x^2 + 20x + 25 = 49 \)[/tex].
- Not a solution.
C. [tex]\( x = \frac{\sqrt{7}-5}{2} \)[/tex]
- Substitute [tex]\( x = \frac{\sqrt{7}-5}{2} \)[/tex]:
[tex]\[ \text{This value does not satisfy the simplified quadratic equation.} \][/tex]
- Not a solution.
D. [tex]\( x = \frac{-\sqrt{7}-5}{2} \)[/tex]
- Substitute [tex]\( x = \frac{-\sqrt{7}-5}{2} \)[/tex]:
[tex]\[ \text{This value does not satisfy the simplified quadratic equation.} \][/tex]
- Not a solution.
E. [tex]\( x = 1 \)[/tex]
- Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[ 4(1)^2 + 20(1) + 25 = 4 + 20 + 25 = 49 \][/tex]
This does satisfy the equation.
- This is a solution.
F. [tex]\( x = 6 \)[/tex]
- Substitute [tex]\( x = 6 \)[/tex]:
[tex]\[ 4(6)^2 + 20(6) + 25 = 289 \][/tex]
This does not satisfy the equation.
- Not a solution.
Therefore, the solutions that apply are:
- [tex]\( x = 1 \)[/tex] (Option E)
