Solve
1. 3x^2-9x=0
2. 12x^2+30x=-12
3. 9a(a+1)=4
4. 16x^2=25
Set each equation equal to 0.
Try to factor.
Set each factor equal to 0.
Solve for x.
I'll pick one of your problems to show you how to solve.
Let's try problem #2:
12x^2 + 30x = -12
Set the equation equal to 0:
12x^2 + 30x + 12 = 0
Try to factor:
(6x + 3)(2x + 4) = 0
Set each factor equal to 0:
6x + 3 = 0
2x + 4 = 0
I'll let you finish this to solve for x.
(Hint: there will be two possible solutions.)
Try this technique with your other problems.
I hope this will help get you started.
To solve these quadratic equations, we can follow a general process called factoring, completing the square, or using the quadratic formula. Let's go through each equation step by step.
1. 3x^2 - 9x = 0
First, we factor out the greatest common factor, which is 3x:
3x(x - 3) = 0
Now, we can set each factor equal to zero and solve for x:
3x = 0 ---> x = 0
x - 3 = 0 ---> x = 3
Therefore, the solutions for this equation are x = 0 and x = 3.
2. 12x^2 + 30x = -12
We start by rewriting the equation in standard form with zero on one side:
12x^2 + 30x + 12 = 0
Next, we can factor out a common factor of 6:
6(2x^2 + 5x + 2) = 0
Then, we can factor the quadratic expression inside the parentheses:
6(2x + 1)(x + 2) = 0
Setting each factor equal to zero, we can solve for x:
2x + 1 = 0 ---> x = -1/2
x + 2 = 0 ---> x = -2
The solutions to this equation are x = -1/2 and x = -2.
3. 9a(a + 1) = 4
First, we expand the equation:
9a^2 + 9a = 4
Rearranging the equation brings it to standard quadratic form:
9a^2 + 9a - 4 = 0
This equation cannot be easily factored, so let's use the quadratic formula to find the solutions:
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 9, b = 9, and c = -4. Plugging these values into the quadratic formula, we get:
a = 9, b = 9, c = -4
a = 9, b = 9, c = -4
Using the quadratic formula:
a = 9, b = 9, c = -4
x = (-9 ± √(9^2 - 4 * 9 * -4)) / (2 * 9)
Simplifying further:
x = (-9 ± √(81 + 144)) / 18
x = (-9 ± √(225)) / 18
x = (-9 ± 15) / 18
x = (-9 + 15) / 18 ---> x = 6 / 18 ---> x = 1/3
x = (-9 - 15) / 18 ---> x = -24 / 18 ---> x = -4/3
Thus, the solutions to this equation are x = 1/3 and x = -4/3.
4. 16x^2 = 25
First, we bring the equation to standard quadratic form by rewriting it with zero on one side:
16x^2 - 25 = 0
This equation can be factored using the difference of squares formula:
(4x - 5)(4x + 5) = 0
Setting each factor equal to zero, we can solve for x:
4x - 5 = 0 ---> 4x = 5 ---> x = 5/4
4x + 5 = 0 ---> 4x = -5 ---> x = -5/4
The solutions for this equation are x = 5/4 and x = -5/4.