Answer:
1. x = 3, 1/2 (solutions); (x - 3)(2x - 1) (factors)
2. x = -1/3, -2 (solutions); (3x + 1)(x + 2) (factors)
Step-by-step explanation:
<u>1. 2x^2 - 7x + 3</u>
To solve problem 1, you will need to identify your a, b, and c values in this quadratic function.
Since this problem is in standard form, it will be easy to identify these values. The standard form of a quadratic function is ax^2 + bx + c.
The a value is 2, the b value is -7, and the c value is 3 if we use our standard form and see which numbers are plugged into it.
Since we know that
- a = 2
- b = -7
- c = 3
we can use the quadratic formula:
Substitute the a, b, and c values into the quadratic formula:
Now simplify using the laws of pemdas:
Simplify even further:
Now split this equation into two equations to solve for x:
12/4 can be simplified to 3, and 2/4 can be simplified to 1/2.
This means your solutions to problem 1 is 3, 1/2.
There is also another way to solve for the quadratic functions, and this was by factoring.
If you factor 2x^2 - 7x + 3 using the bottoms-up method, you will get (x - 3)(2x - 1).
After factoring, solving for the solutions is simple because all you have to do is set each factor to 0.
- x - 3 = 0
- 2x - 1 = 0
After solving for x by adding 3 to both sides, or by adding 1 to both sides then dividing by 2, you will end up with the same solutions: x = 3 and x = 1/2.
<u>2. 3x^2 + 7x + 2</u>
To save time I'll be using the bottoms-up factoring method, but remember to refer back to problem 1 (quadratic formula) if you prefer that method.
Factor this quadratic function using the bottoms-up method. After factoring you will have (3x + 1)(x + 2). These are your factors.
Now to solve for x and find the solutions of the quadratic function, you will set both factors equal to 0.
- 3x + 1 = 0
- x + 2 = 0
Solve.
<u>First factor:</u> 3x + 1 = 0
Subtract 1 from both sides.
3x = -1
Divide both sides by 3.
x = -1/3
<u>Second factor:</u> x + 2 = 0
Subtract 2 from both sides.
x = -2
Your solutions are x = -1/3 and x = -2.