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Question

B=3abx²-5x solve for x

p(x) = -x² - 8x - 12 find vertex, Y intercept and X interxept and graph.

Answer 1

1).

The first function is B = 3abx ^2 - 5x.

3abx ^2 - 5x - B = 0.

Compare the above equation with quadratic equation ax ^2 + bx + c = 0.

Solution x = [- b ± √(b ^2 - 4ac )] / 2a.

Substitute the values b = - 5, a = 3ab, and c = - B.

x = [- (- 5) ± √((- 5)^2 - 4(3ab )(- B))] / 2(3ab )

x = [ 5 ± √(25 + 12abB )] / 6ab.

Therefore, solution of the given function is x = [ 5 + √(25 + 12abB )] / 6ab and [ 5 - √(25 + 12abB)] / 6ab for ab not equals to zero.

2).

P (x ) = - x ^2 - 8x - 12.

Let, P (x ) = y = - x ^2 - 8x - 12.

The above equation represent the parabola.

Here x ² coefficient is - 1, for perfect square make x ² coefficient 1 by multiplying each side by - 1.

- y = x ^2 + 8x + 12.

To change the expression x ^2 + 8x + 12 into a perfect square trinomial add and subtract (half the x coefficient)² to each side of the expression.

Here x coefficient = 8. so, (half the x coefficient)² = (8/2)2= 16.

Add and subtract 16 to the expression.

- y = x  ^2 + 8x + 16 - 16 + 12

- y = (x + 4)^2 - 4

y = - (x + 4)^2 + 4.

Compare the equation y = - (x + 4)^2 + 4 with The standard form of the parabola with vertex (h, k ) and axis of symmetry x = h  is y = a (x - h )2 + k.

Vertex (h, k ) = (- 4, 4), and axis of symmetry h = - 4.

To find the y - intercept, substitute x = 0 in the y = - x ^2 - 8x - 12.

y = - (0)^2 - 8(0) - 12

  = 0 - 0 - 12 = - 12.

y = - 12.

To find the x - intercept, substitute y = 0 in the y = - x ^2 - 8x - 12.

0 = - x ^2 - 8x - 12

0 = x ^2 + 8x + 12

0 = x ^2 + 6x + 2x + 12

0 = x(x + 6) + 2(x + 6)

0 = (x + 6)(x + 2)

x + 6 = 0 and x + 2 = 0

x = - 6 and x = - 2.

Answer 2

Make the table of values to find ordered pairs that satisfy the equation.

Choose values for x and find the corresponding values for y.

x	y = - x ^2 - 8x - 12	(x, y )
- 6	y = - (- 6)^2 - 8(- 6) - 12 = - 36 + 48 - 12 = 48 - 48 = 0	(- 6, 0)
- 5	y = - (- 5)^2 - 8(- 5) - 12 = - 25 + 40 - 12 = 40 - 37 = 3	(- 5, 3)
- 4	y = - (- 4)^2 - 8(- 4) - 12 = - 16 + 32 - 12 = 32 - 28 = 4	(- 4, 4)
- 3	y = - (- 3)^2 - 8(- 3) - 12 = - 9 + 24 - 12 = 24 - 21 = 3	(- 3, 3)
- 2	y = - (- 2)^2 - 8(- 2) - 12 = - 4 + 16 - 12 = 16 - 16 = 0	(- 2, 0)
- 1	y = - (- 1)^2 - 8(- 1) - 12 = -1 + 8 - 12 = 7 - 12 = - 5	(- 1, - 5)

To draw the equation y = - (x + 4)^2 + 4 follow the steps.

1. Draw a coordinate plane.

2. Plot the points on the coordinate plane.

3.Then sketch the graph, connecting the points with a smooth curve.

The points (- 6, 0), (- 5, 3), (- 4, 4), (- 3, 3),(- 2, 0), and (- 1, - 5) are also on the parabola.