How do I solve the following problems??
1. Factor: 4x^2-100y^2
2. Factor: 3x^2+27
3. Solve 3x^2+9x=0
4. Solve 12^2+30x=-12
5. 9a(a+1)=4
6. Solve 16x^2=25
Hints:
1. Factor: 4x^2-100y^2
use difference of two squares,
(a²-b²)=(a+b)(a-b)
2. Factor: 3x^2+27
the common factor is 3.
3. Solve 3x^2+9x=0
factor and solve:
3x(x+3)=0
therefore
x=0 or x=?
4. Solve 12^2+30x=-12
Factor out the 6, and then factor completely to solve for x:
6(2x²+5x+2)=0
Hint: it's in the form
(2x 1)(x 2)
5. 9a(a+1)=4
Expand, transpose everything to the left, and then factor.
9a²+9a-4=0
6. Solve 16x^2=25
isolate variable:
x²=(25/16)=(5/4)²
and solve for x.
To solve these problems, let's go through them step by step:
1. Factor: 4x^2 - 100y^2
To factor this expression, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In this case, a = 2x and b = 10y.
So, we can rewrite the expression as: (2x)^2 - (10y)^2
Now, apply the difference of squares formula: (2x + 10y)(2x - 10y)
Therefore, the expression 4x^2 - 100y^2 can be factored as (2x + 10y)(2x - 10y).
2. Factor: 3x^2 + 27
In this problem, we need to factor out the greatest common factor (GCF), which is 3. Divide each term by 3:
3(x^2 + 9)
Now, we have x^2 + 9, which cannot be factored further because it is a sum of squares. Therefore, the factored form of 3x^2 + 27 is 3(x^2 + 9).
3. Solve 3x^2 + 9x = 0
To solve this quadratic equation, we need to set the equation equal to zero and then factor it:
3x(x + 3) = 0
Now, we have two solutions:
1) 3x = 0, which means x = 0
2) x + 3 = 0, which means x = -3
Therefore, the solutions to 3x^2 + 9x = 0 are x = 0 and x = -3.
4. Solve 12x^2 + 30x = -12
Start by setting the equation equal to zero:
12x^2 + 30x + 12 = 0
We can simplify the equation by dividing each term by 6:
2x^2 + 5x + 2 = 0
Now, we need to factor this quadratic equation. It can be factored as:
(2x + 1)(x + 2) = 0
Setting each factor equal to zero, we have:
1) 2x + 1 = 0, which means x = -1/2
2) x + 2 = 0, which means x = -2
Therefore, the solutions to 12x^2 + 30x = -12 are x = -1/2 and x = -2.
5. 9a(a + 1) = 4
Start by expanding the left side of the equation:
9a^2 + 9a = 4
Rearrange the equation and set it equal to zero:
9a^2 + 9a - 4 = 0
Now, we need to factor this quadratic equation. However, it cannot be factored easily, so we will use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values for a, b, and c:
a = (-9 ± √(9^2 - 4(9)(-4))) / 2(9)
a = (-9 ± √(81 + 144)) / 18
a = (-9 ± √(225)) / 18
The square root of 225 is 15:
a = (-9 ± 15) / 18
Now, we have two possible solutions:
1) (-9 + 15) / 18 = 6 / 18 = 1/3
2) (-9 - 15) / 18 = -24 / 18 = -4/3
Therefore, the solutions to 9a(a + 1) = 4 are a = 1/3 and a = -4/3.
6. Solve 16x^2 = 25
First, rearrange the equation and set it equal to zero:
16x^2 - 25 = 0
Now, we can apply the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In this case, a = 4x and b = 5.
So, we can rewrite the equation as: (4x + 5)(4x - 5) = 0
Setting each factor equal to zero, we have:
1) 4x + 5 = 0, which means x = -5/4
2) 4x - 5 = 0, which means x = 5/4
Therefore, the solutions to 16x^2 = 25 are x = -5/4 and x = 5/4.