Discriminat
And Roots
And Roots
What is a discriminant ?
The discriminant is the part of the quadratic formula under the square root, b^2 - 4ac, and it tells us how many and what type of roots a quadratic equation has.
Factor:
X^2 + 5x + 6
(x + 2)(x + 3)
This is the value you add to both sides to complete the square for x^2 + 6x =__
What is 9
Factor completely
X2+7x+10
(X+2)(X+5)
The area of a square is 49 square meters. What is the length of one side?
x=7
Use the discriminant to determine the nature of the roots for the quadratic equation
-2x2+3x +8 =0
b2 - 4ac = 73
Since the value of the discriminant is positive, there are two distinct real roots.
Factor:
x^2 - 4x - 12
(x - 6)(x + 2)
This is the completed square form of the equation
x^2 + 8x + 16
What is (x+4)(x+4)
Solve this factored equation
2(X-4)(7-2X)
X=4, X=7/2
A number multiplied by 3 more than itself is 40. What is the number?
x= 5, x= -8
Use the discriminant to determine the nature of the roots for the quadratic equation
3x2-5x=-9
b2- 4ac = -83
Since the value of the discriminant is negative, there are no real roots.
The square root of a negative number does not result in a real number.
Factor:
2x^2 + 9x + 4
2x^2 + 8x + x + 4
= 2x(x + 4) + 1(x + 4)
= (2x + 1)(x + 4)
To solve x^2 - 10x +25 = 0 by completing the square, this is the solution
What is x=5?
Solve this quadratic equation by factoring
3w2+28w+9=0
w=-9, x=-1/3
The length of a rectangle is 2 meters more than its width. The area is 48 square meters. What is the width?
x= 6 (width), x+2=8 (length)
Use the discriminant to determine the nature of the roots for the quadratic equation
1/4x2-3x+9=0
b2 - 4ac = 0
Since the value of the discriminant is zero, there is one distinct real root, or two equal real roots.
Factor:
6x^2 + 12x + 6
6(x^2 + 2x + 1)
= 6(x + 1)(x + 1)
= 6(x + 1)^2
This is the completed square form of x^2 +4x +4
(x+2)^2
A ball is thrown upward from the ground with the equation h=−5t^2 +20t, where h is the height in meters after tt seconds. When does it hit the ground?
t= 0s, t= 4s
Use the discriminant to determine the nature of the roots for the quadratic equation
3x2- x - 4 = 0
b2- 4ac = 49
Since the value of the discriminant is positive, there are two distinct real roots.
Factor:
6x^2 + 11x + 3
6x^2 + 9x + 2x + 3
= 3x(2x + 3) + 1(2x + 3)
= (3x + 1)(2x + 3)
Solve 3^2+12x−6=0 by completing the square
x =-2 +/- √6
A company’s profit PP in thousands is given by P(x)=−2x^2+12x−16, where x is the number of items produced in hundreds. How many items should be produced for zero profit?
x=2, x=4 -> 200, 400 items