Factor these:?

Question

Factor: 2x^2-5x-3

Will the following function have a maximum or a minimum? Explain.

f(x)=-4x^2+3x+11

Factor:
50x^2-32

Answer 1

Factor: 2x^2-5x-3
2x^2-5x-3
= 2x^2-6x+x-3
= 2x(x-3) + (x-3)
= (2x+1)(x-3)

Will the following function have a maximum or a minimum? Explain. f(x)=-4x^2+3x+11

Maximum because f(x) opens down.

Factor: 50x^2-32
50x^2-32
= 2(25x^2-16)
= 2(5x+4)(5x-4)

Answer 2

Quadratic equations, like your example are parabolic in shape. They will always have a minimum if the first term has a positive sign, like yours. The shape of the parabola will be curving upward. A maximum if the first term has a negative sign, because the curve goes downward. Draw it and it is easier to see.

In the example f(x)=-4x^2+3x+11, the first term is negative, therefore it will have a Maximum. To factor, factor out the -1 first as -1(4x^2 -3x -11).
To find the maximum, use formula x = (-b/(2a)) from the original equation coefficients. Or,
-3/(2x-4) = -3/-8 = 3/8. Use this formula if you have not studied calculus (derivatives) yet.


Now, factoring, either by inspection, or using GCFs or grouping, you get,

2x^2-5x-3
= 2x^2-6x+x-3
= 2x(x-3) + (x-3)
= (2x+1)(x-3)

Factors of 2x^2 are 2x and x. Factors of 3 are 3 and 1. Combination of these factors must give you the middle term of -5. Note that (2x -3)(x+1) will not work because even though it will give you a negative middle term ( -x) it is not the right one, (-5).


In your last example, factor out 2 first because the terms are even. You will notice that the other factor is of the special form of x^2-y^2. The factors are (x-y)(x+y). So,

Factor: 50x^2-32
50x^2-32
= 2(25x^2-16) = or rewrite as (5x)^2 - (4)^2
= 2(5x-4)(5x+4)

Answer 3

2x^2-5x-3=(2x+1)(x-3)

Try to graph the function. You will find that the function is a parabola opening downward. That is, you have a maximum value at the vertex of the parabola and no minimum value.

50x^2-32=2(25x^2-16)=2(5x-4)(5x+4)

Answer 4

Question 1
(2x + 1).(x - 3)

Question 2
f `(x) = - 8x + 3 = 0 for turning point
8x = 3
x = 3/8
f "(x) = - 8
f "(x) is -ve therefore there will be a Maximum tuning point when x = 3/8
Turning point is (3/8,f(3/8)) and f(3/8) may be calculated if desired.

Question 3
= 2.(25x² - 16)
= 2.(5x - 4).(5x + 4)

Answer 5

2x^2-6x+x-3
=(2x+1)(x-3)

Minimum, coeff. of highest degree is +ve.

2(25x^2-16)
= 2(5x-4)(5x+4)