Which graph can be used to find the solution(s) to 3-4x=-3-x2+4x
Answers
Answer:
1. Consider the functions f(x)=-4x+3 and g(x)=- x^{2} +4x-3−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)−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-3−4x+3=−x
2
+4x−3
0=- x^{2} +8x-60=−x
2
+8x−6
x^{2} -8x+6=0x
2
−8x+6=0
x^{2} -8x+6=x^{2} -2*4x+16-16+6=(x-4)^{2}-10=0x
2
−8x+6=x
2
−2∗4x+16−16+6=(x−4)
2
−10=0
(x-4)^{2}=10(x−4)
2
=10
x-4= +\sqrt{10}x−4=+
10
or x-4= -\sqrt{10}x−4=−
10
x=4 +\sqrt{10}x=4+
10
or x=4 -\sqrt{10}x=4−
10