Which of the following equations has no solution? 73 - 32 - 4= 4
072-3x + 4 = 4
3(2x - 4) = 6 (2-2)
3(2x - 4) = 6 (2+2)

1. Simplify: (4x2 - 2x) - (-5x2 - 8x).

Solution:

(4x2 - 2x) - (-5x2 - 8x)

= 4x2 - 2x + 5x2 + 8x.

= 4x2 + 5x2 - 2x + 8x.

= 9x2 + 6x.

= 3x(3x + 2).

3x(3x + 2)

2. Simplify: (-7x2 - 3x) - (4x2 - x)

Solution:

(-7x2 - 3x) - (4x2 - x)

= -7x2 - 3x - 4x2 + x

= -7x2 - 4x2 - 3x + x

= -11x2 – 2x

= -x(11x + 2)

-x(11x + 2)

3. Simplify: (-5x2 - 8x + 7) - (5x2 - 4)

Solution:

(-5x2 - 8x + 7) - (5x2 - 4)

= -5x2 - 8x + 7 - 5x2 + 4

= -5x2 - 5x2 - 8x + 7 + 4

= -10x2 – 8x + 11

-10x2 – 8x + 11

4. Simplify: (4x2 + 5x) + (x2 - 5x + 1)

Solution:

(4x2 + 5x) + (x2 - 5x + 1)

= 4x2 + 5x + x2 - 5x + 1

= 4x2 + x2+ 5x - 5x + 1

= 5x2 + 0 + 1

= 5x2 + 1

5x2 + 1

5. Simplify: (-6x + 3) - (-x2 + 5x)

Solution:

(-6x + 3) - (-x2 + 5x)

= -6x + 3 + x2 - 5x

= -6x - 5x + x2 + 3

= x2 – 11x + 3

x2 – 11x + 3

6. Simplify: (-9x - 5) - (-9x2 + x - 5)

Solution:

(-9x - 5) - (-9x2 + x - 5)

= -9x - 5 + 9x2 - x + 5

= 9x2 - 9x – x – 5 + 5

= 9x2 – 10x

= x(9x – 10)

x(9x – 10)

7. Simplify: (2x - 4) - (6x + 6)

Solution:

(2x - 4) - (6x + 6)

= 2x - 4 - 6x – 6

= 2x – 6x – 4 – 6

= -4x – 10

= -2(2x + 5)

-2(2x + 5)

8. Simplify:(8+4x)/4x

Solution:

(8+4x)/4x

= 4(2 + x)/4x

[Cancel 4 from the numerator and denominator]

= (2 + x)/x

(2 + x)/x

9. Decide whether the lines are parallel, perpendicular or neither.

x + 4y = 7 and 4x – y = 3

Solution:

x + 4y = 7

4y = -x + 7

y = (-1/4) x + 7

Slope of the equation x + 4y = 7 is -1/4.

Again, 4x – y = 3

y = 4x – 3

Slope of the equation 4x – y = 3 is 4.

Since multiplying both the slope of the equation = -1/4 × 4 = -1

Therefore, the given two equations are perpendicular to each other.

Perpendicular

10. Simplify: 3x – 5 + 23x – 9

Solution:

3x – 5 + 23x – 9

= 3x + 23x – 5 – 9

= 26x – 14

= 2(13x – 7)

2(13x – 7)

11. Use the discriminant to determine the number of real roots the equation has. 3x2 – 5x + 1 =0

(a) One real root (a double root),

(b) Two distinct real roots,

(c) Three real roots,

(d) None (two imaginary roots)

Solution:

Discriminant = bx2 – 4ac

Compare the above equation 3x2 – 5x + 1 =0 with ax2 + bx + c = 0

We get, a = 3, b = -5, c = 1

Put the value of a, b and c;

Discriminant = bx2 – 4ac

Discriminant = (-5)2 - 4 × 3 × 1

= 25 – 12

= 13 [13 > 0]

Therefore, discriminant is positive.

So the given equation has two distinct real roots.

(b)

12. Use the discriminant to determine the number of real roots the equation has. 7x2 + 3x + 1 =0.

(a) One real root (a double root),

(b) Two distinct real roots,

(c) Three real roots,

(d) None (two imaginary roots)

Solution:

Discriminant = b2 – 4ac

Compare the above equation 7x2 + 3x + 1 =0 with ax2 + bx + c = 0

We get, a = 7, b = 3, c = 1

Put the value of a, b and c;

Discriminant = b2 – 4ac

Discriminant = (3)2 - 4 × 7 × 1

= 9 – 28

= -19 [13 < 0]

Therefore, discriminant is negative.

So the given equation has none (two imaginary roots).

(d)

13. Fill in the blank:

(a) The point with coordinates (0,0) is called of a rectangular coordinate system.

(b) To find the x-intercept of a line, we letequal 0 and solve for ; to find y-intercept, we let equal 0 and solve for

Solution:

(a) The point with coordinates (0,0) is called origin of a rectangular coordinate system.

(b) To find the x-intercept of a line, we let y equal 0 and solve for x ; to find y-intercept, we let x equal 0 and solve for y .

14. Name the quadrant, if any, in which each point is located.

(a) (1, 6)

(b) (-4, -2)

Solution:

(a) (1, 6) I Quadrant.

(b) (-4, -2) III Quadrant.

arithmetic sequence

step-by-step explanation:

given that a sequence has terms as

19, 15, 11,

this is represented on a graph

the pattern indicates that the sequence is an arithmetic sequence with i term

a=19 and common difference d =-4

any general term

thus the sequence is reduced by 4 for each successive term starting from 1 and nth term is

24-5n.

first negative term would start after 5th term

5th term =24-25< 0

from 5th term onwards the sequence would be negative.