factorise: 2x^2+6x-16, and 14x^2-3x-2.
Solve: 6y^2-5y-6=0, and 8x^2-2x-1=0.
Rearrange: p=5t-u/u to make u the subject, and p(q+r)=(q-p).
please help i've done about 40 questions and im just stuck on these. thanks for the help
2x^2+6x-16
= 2(x^2 + 3x - 8)
14x^2-3x-2
= (2x - 1)(7x + 2)
6y^2 - 5y - 6 = 0
(3y + 2 )(2y - 3 ) = 0
y = - 2/3 or y = 3/2
8x^2-2x-1=0
(4x + 1)(2x - 1) = 0
x = - 1/4 or x = 1/2
p=5t-u/u
I think you mean p=(5t-u)/u or else the u's would cancel
up = 5t - u
up + u = 5t
u(p+1) = 5t
u = 5t/(p+1)
p(q+r) = (q-p)
you don't say which variable to solve for, but expand, bring the necessary variable to the left, factor it out like I did in the previous problem
thanks for the help but cant
2x^2+6x-16
= 2(x^2 + 3x - 8)
go any further
ps thanks Reiny
I'll be happy to help you with the questions you have.
1. Factorize: 2x^2 + 6x - 16
To factorize this quadratic expression, we need to find two binomials such that when multiplied, they result in the given expression.
First, we look for two numbers whose product is equal to the product of the coefficient of the first term (2) and the constant term (-16). In this case, the product is -32.
Next, we need to find two numbers whose sum is equal to the coefficient of the second term (+6). In this case, the sum is 6.
By trial and error, we can determine that the two numbers are +8 and -4.
Therefore, the factorization of the expression is: (2x - 4)(x + 8).
2. Factorize: 14x^2 - 3x - 2
Following a similar process, we look for two numbers whose product is equal to the product of the coefficient of the first term (14) and the constant term (-2). In this case, the product is -28.
Next, we need to find two numbers whose sum is equal to the coefficient of the second term (-3). In this case, the sum is -3.
By trial and error, we can determine that the two numbers are -7 and +4.
Therefore, the factorization of the expression is: (2x - 1)(7x + 2).
3. Solve: 6y^2 - 5y - 6 = 0
To solve this quadratic equation, we can use either factoring, the quadratic formula, or completing the square.
In this case, factoring is the most efficient method. We need to find two binomials such that when multiplied, they yield the given equation.
The factors of 6y^2 are (2y)(3y) or (6y)(y).
The factors of -6 are (-1)(6) or (1)(-6).
We can rearrange the equation by replacing -5y with the factors of -6 that will give -5y:
6y^2 - 6y + y - 6 = 0
Grouping the terms:
(6y^2 - 6y) + (y - 6) = 0
Factor by grouping:
6y(y - 1) + (y - 1) = 0
Now we can factor out the common binomial:
(y - 1)(6y + 1) = 0
From here, we set each factor equal to zero and solve for y.
y - 1 = 0 => y = 1
6y + 1 = 0 => 6y = -1 => y = -1/6
Therefore, the solutions to the equation are y = 1 and y = -1/6.
4. Solve: 8x^2 - 2x - 1 = 0
Similarly to the previous equation, we can use factoring, the quadratic formula, or completing the square.
In this case, we'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
Given the equation: 8x^2 - 2x - 1 = 0
From the equation, we can identify a = 8, b = -2, and c = -1.
Substituting these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4(8)(-1))) / (2(8))
x = (2 ± √(4 + 32)) / 16
x = (2 ± √36) / 16
x = (2 ± 6) / 16
This gives us two solutions:
x = (2 + 6) / 16 => x = 8 / 16 => x = 1/2
x = (2 - 6) / 16 => x = -4 / 16 => x = -1/4
Therefore, the solutions to the equation are x = 1/2 and x = -1/4.
5. Rearrange: p = (5t - u) / u to make u the subject
To make u the subject, we first multiply both sides of the equation by u:
u * p = 5t - u
Next, we can rearrange the equation, isolating the term with u on the left side:
u + u * p = 5t
Factoring out u:
u(1 + p) = 5t
Finally, divide both sides by (1 + p) to solve for u:
u = 5t / (1 + p)
6. Rearrange: p(q + r) = (q - p) to solve for p
To solve for p, we need to isolate it on one side of the equation.
First, expand the left side of the equation:
pq + pr = q - p
Next, bring all the terms with p to one side of the equation:
pq + p = q - pr
Factor out p from the left side of the equation:
p(q + 1) = q - pr
Finally, divide both sides by (q + 1) to solve for p:
p = (q - pr) / (q + 1)
I hope this helps! Let me know if you have any further questions or need any additional explanations.