1. What is the sum and product of the roots of the quadratic equation [tex] 4m^2 - 8m + 8 = 0 [/tex]?
Solution:
Sum of the roots = [tex] \displaystyle -b/a = -(-8)/4 = 8/4 = \boxed{2} [/tex]
Product of the roots = [tex] \displaystyle c/a = 8/4 = \boxed{2} [/tex]
Unfortunately there are no real roots exists for the equation [tex] 4m^2 - 8m + 8 = 0 [/tex]. So it this still valid? Yes. Turns out that their complex solution has a sum of 2 and a product of 2.
2. Determine the quadratic equation in standard form given the roots [tex] (5,0) [/tex]
Solution:
[tex] \displaystyle x = -5, x = 0 [/tex]
[tex] \displaystyle x + 5 = 0, x = 0 [/tex]
[tex] \displaystyle (x + 5)x [/tex]
[tex] \displaystyle \boxed{x^2 + 5x = 0} [/tex]
3. What do you call an expression that have a polynomial/s in the numerator or denominator?
[tex]\displaystyle \textsf{Rational algebraic expression}[/tex]
4. A rectangular garden has an area of 14m² and a perimeter of 18 meters. find the dimensions of the rectangular garden.
Solution:
[tex] \displaystyle l \cdot w = 14, 2l + 2w = 18 [/tex]
Find l:
[tex] \displaystyle 2l + 2w = 18 [/tex]
[tex] \displaystyle 2l = 18 - 2w [/tex]
[tex] \displaystyle l = 9 - w [/tex]
Replace l with w:
[tex] \displaystyle l \cdot w = 14 [/tex]
[tex] \displaystyle (9 - w) \cdot w = 14 [/tex]
[tex] \displaystyle 9w - w^2 = 14 [/tex]
[tex] \displaystyle 9w - w^2 - 14 = 0 [/tex]
Divide both sides by negative 1:
[tex] \displaystyle -9w + w^2 + 14 = 0 [/tex]
Arrange:
[tex] \displaystyle \boxed{ w^2 - 9w + 14 = 0 } [/tex]
5. The product of two consecutive odd integers is 99. Find the integers.
Solution:
Let [tex] x, x+2 [/tex] be the odd consecutive integers, then:
[tex] \displaystyle x(x+2) = 99 [/tex]
[tex] \displaystyle x^2 + 2x = 99 [/tex]
[tex] \displaystyle x^2 + 2x - 99 = 0 [/tex]
Solve by factoring:
[tex] \displaystyle ( x + 11 )( x - 9) [/tex]
[tex]\displaystyle \boxed{ x = -11, 9 }[/tex]
For the other one:
Let [tex] x, x-2 [/tex] be the odd consecutive integers, then:
[tex] \displaystyle x(x-2) = 99 [/tex]
[tex] \displaystyle x^2 - 2x = 99 [/tex]
[tex] \displaystyle x^2 - 2x - 99 = 0 [/tex]
Solve by factoring:
[tex] \displaystyle ( x - 11 )( x + 9) [/tex]
[tex]\displaystyle \boxed{ x = 11, -9 }[/tex]
6. The product of two consecutive even integers is 168. find the integers.
Solution:
Let [tex] x, x+2 [/tex] be the even consecutive integers, then:
[tex] \displaystyle x(x+2) = 168 [/tex]
[tex] \displaystyle x^2 + 2x = 168 [/tex]
[tex] \displaystyle x^2 + 2x - 168 = 0 [/tex]
Solve by factoring:
[tex] \displaystyle ( x + 12 )( x - 14) [/tex]
[tex]\displaystyle \boxed{ x = -12, 14 }[/tex]
For the other one:
Let [tex] x, x-2 [/tex] be the even consecutive integers, then:
[tex] \displaystyle x(x-2) = 168 [/tex]
[tex] \displaystyle x^2 - 2x = 168 [/tex]
[tex] \displaystyle x^2 - 2x - 168 = 0 [/tex]
Solve by factoring:
[tex] \displaystyle ( x - 12 )( x + 14) [/tex]
[tex]\displaystyle \boxed{ x = 12, -14 }[/tex]
7. Find for the solutions of the equation [tex] x(x - 5) = 36 [/tex]
Solution:
[tex]\displaystyle x (x-5) = 36[/tex]
[tex] \displaystyle x^2 - 5x = 36 [/tex]
[tex] \displaystyle x^2 - 5x - 36 = 0 [/tex]
[tex] \displaystyle ( x + 4 )( x - 9 ) = 0 [/tex]
[tex] \displaystyle \boxed{ x = -4, 9} [/tex]
8. Which of the following equation is an example of a quadratic inequality?
[tex] \displaystyle x^2 - 4x + 4 > 0 [/tex]
9. If the value of x in the quadratic inequality is greater than or greater than or equal to a certain number then the solutions is?
[tex]\displaystyle \textsf{Opposite directions on the number line}[/tex]
10. If the value of x in the quadratic inequality is less than or less than or equal to a certain number then the solutions is?
[tex]\displaystyle \textsf{Toward each other on the number line}[/tex]
1. [ Not in the choices ]
2. D
3. A
4. A
5. D
6. B
7. B
8. A
9. C
10. A
