Determine the points of intersection of the line y = -2x + 7 and the parabola y = 2x2 + 3x - 5.
1 answer:
Answer:
Step-by-step explanation:
The x-values of those points can be found by equating the expressions for y, then solving for x.
... -2x +7 = 2x² +3x -5
... 2x² +5x -12 = 0 . . . . . . . . subtract the left side to put into standard form
This can be factored by looking fro two factors of 2·(-12) = -24 that add to give 5, the x-coefficient. Those factors are 8, -3.
... 2x² +8x -3x -12 = 0 . . . . rewrite the middle term
... 2x(x +4) -3(x +4) = 0 . . . factor by pairs
... (x +4)(2x -3) = 0 . . . . . . . complete factorization
... x = -4, x = 3/2 . . . . . . . . values of x that make the factors be zero
The corresponding y-values are ...
... for x = -4, y = -2(-4) +7 = 15
... for x = 3/2, y = -2(3/2) +7 = 4
The points of intersection are (-4, 15) and (3/2, 4).
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ANSWER
EXPLANATION.
The orthocenter is the point of intersection of any two altitudes of the triangle.
First, you need to determine the slope of each side of the triangle.
△ABC has vertices A(2, 3), B(−4, −3), C(2, −3).
This line has undefined slope, which means it is a vertical line.
The slope of this line is zero, meaning the line is a horizontal line.
This implies that side BC and side AC of the given triangle are perpendicular and will intersect at C since they are the altitudes of triangle ABC.
Hence the orthocentre is
See diagram in attachment.
EXPLANATION.
The orthocenter is the point of intersection of any two altitudes of the triangle.
First, you need to determine the slope of each side of the triangle.
△ABC has vertices A(2, 3), B(−4, −3), C(2, −3).
This line has undefined slope, which means it is a vertical line.
The slope of this line is zero, meaning the line is a horizontal line.
This implies that side BC and side AC of the given triangle are perpendicular and will intersect at C since they are the altitudes of triangle ABC.
Hence the orthocentre is
See diagram in attachment.