Question

Which graph can be used to find the solution(s) to 3 – 4x = –3 – x2 + 4x?

Answer 1

1. Consider the functions f(x)=-4x+3 and g(x)=$- x^{2} +4x-3$

the graph of f is a line (f is a linear polynomial function) so we can use 2 points to draw it, say (0, 3) and (2, -5)

the graph of g is a parabola (g is a quadratic polynomial function), so we can factorize the expression as follows: 

$- x^{2} +4x-3=-(x^{2} -4x+3)=-(x-3)(x-1)$

so the roots of g are 1 and 3, the axis of symmetry is the vertical line through (2,0), so the vertex is calculated at x=2, -(2-3)(2-1)=-(-1)(1)=1. Vertex is the point (2, 1)

2. check the picture attached. Clearly there are 2 solutions, at the 2 intersections of the graphs.

3. Algebraically, the solutions are at :

$-4x+3 =- x^{2} +4x-3$

$0=- x^{2} +8x-6$

$x^{2} -8x+6=0$

$x^{2} -8x+6=x^{2} -2*4x+16-16+6=(x-4)^{2}-10=0$

$(x-4)^{2}=10$

$x-4= +\sqrt{10}$ or $x-4= -\sqrt{10}$

$x=4 +\sqrt{10}$ or $x=4 -\sqrt{10}$

Answer 2

Answer:

D

Step-by-step explanation:

graphing calculator