The parabolas defined by the equations y = -x^2 - x + 3 and y = 2x^2 - 1 intersect at points (a, b) and (c, d), where c is greater than or equal to a. What is c - a? Express your answer as a common fraction.
+36916 - x^2 -x + 3 = 2x^2 -1 re-arrange
0 = 3x^2 +x-4 use quadratic formula to find the x values where the graphs intersect
a = 3 b = 1 c = -4
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
use the x values in the equations to calculate the corresponding 'y' values
(actually....you will only need the 'x' values which will be c and a )
...then you can answer the rest of the question
+14917 The parabolas defined by the equations y = -x^2 - x + 3 and y = 2x^2 - 1 intersect at points (a, b) and (c, d), where c is greater than or equal to a. What is c - a? Express your answer as a common fraction.
Hello Guest!
\(y = -x ^ 2 - x + 3\\ y = 2x ^ 2 - 1\\ \color{black}2x^2-1=-x^2-x+3 \)
\(3x^2+x-4=0\)
\(x = {-1 \pm \sqrt{1+4\cdot 3\cdot 4} \over 2\cdot 3}\\ x=\frac{-1\pm 7}{6}\)
\(a=-\frac{4}{3}\\ \color{black}b=\frac{23}{9}=2.5\overline {5}\\ c=1\\ d=1\)
\(c-a=1-(-\frac{4}{3})\)
\(c-a=\frac{7}{3}\)
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