Answer:
![\boxed{\text{B. x = 4 and x = -1.75}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Ctext%7BB.%20x%20%3D%204%20and%20x%20%3D%20-1.75%7D%7D)
Step-by-step explanation:
ƒ(x) = x² - 2x – 8; g(x) = ¼x -1
If ƒ(x) = g(x), then
x² - 2x – 8 = ¼x -1
One way to solve this problem is by completing the square.
Step 1. Subtract ¼ x from each side
![x^{2} - \dfrac{9}{4}x - 8 = -1](https://tex.z-dn.net/?f=x%5E%7B2%7D%20-%20%5Cdfrac%7B9%7D%7B4%7Dx%20-%208%20%3D%20-1)
Step 2. Move the constant term to the other side of the equation
![x^{2} - \dfrac{9}{4}x = 7](https://tex.z-dn.net/?f=x%5E%7B2%7D%20-%20%5Cdfrac%7B9%7D%7B4%7Dx%20%3D%207)
Step 3. Complete the square on the left-hand side
Take half the coefficient of x, square it, and add it to each side of the equation.
![\dfrac{1}{2} \times \dfrac{9}{4} = \dfrac{9}{8};\qquad \left(\dfrac{9}{8}\right)^{2} = \dfrac{81}{64}\\\\x^{2} - \dfrac{9}{4}x + \dfrac{81}{64} = 7\dfrac{81}{64} = \dfrac{529}{64}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20%5Cdfrac%7B9%7D%7B4%7D%20%3D%20%5Cdfrac%7B9%7D%7B8%7D%3B%5Cqquad%20%5Cleft%28%5Cdfrac%7B9%7D%7B8%7D%5Cright%29%5E%7B2%7D%20%3D%20%5Cdfrac%7B81%7D%7B64%7D%5C%5C%5C%5Cx%5E%7B2%7D%20-%20%5Cdfrac%7B9%7D%7B4%7Dx%20%2B%20%5Cdfrac%7B81%7D%7B64%7D%20%3D%207%5Cdfrac%7B81%7D%7B64%7D%20%3D%20%5Cdfrac%7B529%7D%7B64%7D)
Step 4. Write the left-hand side as a perfect square
![\dfrac{1}{2} \times \dfrac{9}{4} = \dfrac{9}{8};\qquad \left(\dfrac{9}{8}\right)^{2} = \dfrac{81}{64}\\\\x^{2} - \dfrac{9}{4}x + \dfrac{81}{64} = 7\dfrac{81}{64} = \dfrac{529}{64}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20%5Cdfrac%7B9%7D%7B4%7D%20%3D%20%5Cdfrac%7B9%7D%7B8%7D%3B%5Cqquad%20%5Cleft%28%5Cdfrac%7B9%7D%7B8%7D%5Cright%29%5E%7B2%7D%20%3D%20%5Cdfrac%7B81%7D%7B64%7D%5C%5C%5C%5Cx%5E%7B2%7D%20-%20%5Cdfrac%7B9%7D%7B4%7Dx%20%2B%20%5Cdfrac%7B81%7D%7B64%7D%20%3D%207%5Cdfrac%7B81%7D%7B64%7D%20%3D%20%5Cdfrac%7B529%7D%7B64%7D)
Step 5. Take the square root of each side
![x - \dfrac{9}{8} = \pm\sqrt{\dfrac{529}{64}} = \pm\dfrac{23}{8}](https://tex.z-dn.net/?f=x%20-%20%5Cdfrac%7B9%7D%7B8%7D%20%3D%20%5Cpm%5Csqrt%7B%5Cdfrac%7B529%7D%7B64%7D%7D%20%3D%20%5Cpm%5Cdfrac%7B23%7D%7B8%7D)
Step 6. Solve for x
![\begin{array}{rlcrl}x - \dfrac{9}{8} & =\dfrac{23}{8}& \qquad & x - \dfrac{9}{8} & = -\dfrac{23}{8} \\\\x & =\dfrac{23}{8} + \dfrac{9}{8}&\qquad & x & = -\dfrac{23}{8} + \dfrac{9}{8} \\\\x& =\dfrac{32}{8} &\qquad & x & \ -\dfrac{14}{8} \\\\x& =4 & \qquad & x & -1.75 \\\end{array}\\\\\text{f(x) = g(x) when \boxed{\textbf{x = 4 or x = -1.75}}}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brlcrl%7Dx%20-%20%5Cdfrac%7B9%7D%7B8%7D%20%26%20%3D%5Cdfrac%7B23%7D%7B8%7D%26%20%5Cqquad%20%26%20x%20-%20%5Cdfrac%7B9%7D%7B8%7D%20%26%20%3D%20-%5Cdfrac%7B23%7D%7B8%7D%20%5C%5C%5C%5Cx%20%26%20%3D%5Cdfrac%7B23%7D%7B8%7D%20%2B%20%5Cdfrac%7B9%7D%7B8%7D%26%5Cqquad%20%26%20x%20%26%20%3D%20-%5Cdfrac%7B23%7D%7B8%7D%20%2B%20%5Cdfrac%7B9%7D%7B8%7D%20%5C%5C%5C%5Cx%26%20%3D%5Cdfrac%7B32%7D%7B8%7D%20%26%5Cqquad%20%26%20x%20%26%20%5C%20-%5Cdfrac%7B14%7D%7B8%7D%20%5C%5C%5C%5Cx%26%20%3D4%20%26%20%5Cqquad%20%26%20x%20%26%20-1.75%20%5C%5C%5Cend%7Barray%7D%5C%5C%5C%5C%5Ctext%7Bf%28x%29%20%3D%20g%28x%29%20when%20%5Cboxed%7B%5Ctextbf%7Bx%20%3D%204%20or%20x%20%3D%20-1.75%7D%7D%7D)
Check:
![\begin{array}{rlcrl}4^{2} - 2(4) - 8 & = \dfrac{1}{4}(4) -1&\qquad & (-1.75)^{2} - 2(-1.75) - 8 & = \dfrac{1}{4}(-1.75) - 1\\\\16 - 8 -8& = 1 - 1&\qquad & 3.0625 +3.5 - 8 & = -0.4375 - 1 \\\\0& =0&\qquad & -1.4375 & = -1.4375 \\\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brlcrl%7D4%5E%7B2%7D%20-%202%284%29%20-%208%20%26%20%3D%20%5Cdfrac%7B1%7D%7B4%7D%284%29%20-1%26%5Cqquad%20%26%20%28-1.75%29%5E%7B2%7D%20-%202%28-1.75%29%20-%208%20%26%20%3D%20%5Cdfrac%7B1%7D%7B4%7D%28-1.75%29%20-%201%5C%5C%5C%5C16%20-%208%20-8%26%20%3D%201%20-%201%26%5Cqquad%20%26%203.0625%20%2B3.5%20-%208%20%26%20%3D%20-0.4375%20-%201%20%5C%5C%5C%5C0%26%20%3D0%26%5Cqquad%20%26%20-1.4375%20%26%20%3D%20-1.4375%20%5C%5C%5C%5C%5Cend%7Barray%7D)
The diagram below shows that the graph of g(x) intersects that of the parabola ƒ(x) at x = -1.7 and x = 4.