Understanding And Solving 3X²+4X+3

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Introduction

As a professional teacher, it's not uncommon for me to come across students who struggle with algebraic expressions. One such expression that can be particularly challenging is 3x²+4x+3. In this article, we'll break down what this expression means, and go through steps to solve it.

What is 3x²+4x+3?

3x²+4x+3 is a quadratic expression, which means it's an equation that contains a variable raised to the power of two. In this case, that variable is x. The expression has three terms: 3x², 4x, and 3. The first term, 3x², is the quadratic term, the second term, 4x, is the linear term, and the third term, 3, is the constant term.

Solving 3x²+4x+3

To solve 3x²+4x+3, we'll need to use a combination of algebraic techniques, including factoring and the quadratic formula.

Step 1: Factor the quadratic term

The first step in solving 3x²+4x+3 is to factor the quadratic term, 3x². We can do this by finding two numbers that multiply together to give us 3x², and also add up to the linear term, 4x. Those numbers are 3x and x. So, we can rewrite the expression as: 3x²+4x+3 = (3x+ )(x+ )

Step 2: Find the missing numbers to complete the factoring

Now that we have the factored form of 3x²+4x+3, we need to find the missing numbers to complete the factoring. We know that the two terms in the parentheses must multiply together to give us the original expression. So, we need to find two numbers that multiply together to give us 3, and also add up to 3. Those numbers are 1 and 3. So, we can rewrite the expression as: 3x²+4x+3 = (3x+1)(x+3)

Step 3: Check the solution

To check that our factoring is correct, we can use the distributive property to expand the factored form back into the original expression. (3x+1)(x+3) = 3x(x+3)+1(x+3) = 3x²+9x+x+3 = 3x²+4x+3 So, our factored form is correct.

Step 4: Solve for x

Now that we have the factored form of 3x²+4x+3, we can solve for x by setting each term in the parentheses equal to zero and solving for x. 3x+1 = 0 x+3 = 0 Solving for x, we get: x = -1/3 x = -3 So, the solutions to 3x²+4x+3 are x=-1/3 and x=-3.

Conclusion

Understanding and solving quadratic expressions like 3x²+4x+3 can be challenging, but with the right techniques, it's possible to simplify and solve them. By factoring the quadratic term, finding the missing numbers to complete the factoring, and solving for x, we can find the solutions to this expression.