Math Problem Statement

Solve the equation by factoring. (Enter your answers as a comma-separated list.) −4x2 − 6x + 4 = 0 slove for x

Solution

The given quadratic equation is:

4x26x+4=0-4x^2 - 6x + 4 = 0

Step 1: Simplify the equation

We can start by dividing the entire equation by 2-2 to simplify it:

2x2+3x2=02x^2 + 3x - 2 = 0

Step 2: Factor the quadratic equation

We are looking for two numbers that multiply to 2×(2)=42 \times (-2) = -4 and add up to 33.

The numbers 44 and 1-1 satisfy these conditions, because:

4×(1)=4and4+(1)=34 \times (-1) = -4 \quad \text{and} \quad 4 + (-1) = 3

Now, we can rewrite the middle term and factor by grouping:

2x2+4xx2=02x^2 + 4x - x - 2 = 0

Group the terms:

(2x2+4x)(x+2)=0(2x^2 + 4x) - (x + 2) = 0

Factor each group:

2x(x+2)1(x+2)=02x(x + 2) - 1(x + 2) = 0

Factor out the common binomial factor (x+2)(x + 2):

(x+2)(2x1)=0(x + 2)(2x - 1) = 0

Step 3: Solve for xx

Set each factor equal to 0:

  1. x+2=0x=2x + 2 = 0 \quad \Rightarrow \quad x = -2
  2. 2x1=0x=122x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2}

Final Answer:

The solutions are:

x=2,12x = -2, \frac{1}{2}

Would you like further details or have any questions?


Here are 5 related questions:

  1. How can you check if your factored solution is correct?
  2. What are other methods to solve a quadratic equation?
  3. What is the quadratic formula, and how does it apply here?
  4. How do you know when a quadratic equation cannot be factored?
  5. What are the differences between factoring and completing the square?

Tip: Always simplify the quadratic equation by removing common factors before factoring.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Quadratic Equations

Formulas

Quadratic equation formula: ax^2 + bx + c = 0
Factoring by grouping

Theorems

Zero Product Property

Suitable Grade Level

Grades 9-11