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.
