Alright, let's solve this by finding the approximate [tex]\( x \)[/tex]-value where the two equations intersect:
The equations are:
[tex]\[ y = \sqrt{1 - x^2} \][/tex]
[tex]\[ y = 2x - 1 \][/tex]
To find the point of intersection, we set the two equations equal to each other:
[tex]\[ \sqrt{1 - x^2} = 2x - 1 \][/tex]
This equation represents where the two curves intersect. Solving for [tex]\( x \)[/tex]:
1. First, square both sides to eliminate the square root:
[tex]\[ (\sqrt{1 - x^2})^2 = (2x - 1)^2 \][/tex]
[tex]\[ 1 - x^2 = (2x - 1)^2 \][/tex]
2. Expand the right-hand side:
[tex]\[ 1 - x^2 = (2x - 1)(2x - 1) \][/tex]
[tex]\[ 1 - x^2 = 4x^2 - 4x + 1 \][/tex]
3. Move all terms to one side to set the equation to zero:
[tex]\[ 1 - x^2 - 4x^2 + 4x - 1 = 0 \][/tex]
[tex]\[ -5x^2 + 4x = 0 \][/tex]
4. Factor out what you can:
[tex]\[ -x(5x - 4) = 0 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[ -x = 0 \quad \text{or} \quad 5x - 4 = 0 \][/tex]
[tex]\[ x = 0 \quad \text{or} \quad x = \frac{4}{5} \][/tex]
6. Evaluate the valid solution within the allowed domain of the original equations. Here, [tex]\( \frac{4}{5} = 0.8 \)[/tex] is the value that fits the approximate [tex]\( x \)[/tex]-value where the curves intersect.
Thus, the correct answer is:
[tex]\[ \boxed{0.8} \][/tex]
