To solve the quadratic equation [tex]\(3x^2 - 7x + 2 = 0\)[/tex] using the factorization method, we follow these steps:
1. Multiply the coefficient of [tex]\(x^2\)[/tex] (which is 3) by the constant term (which is 2):
[tex]\[3 \times 2 = 6.\][/tex]
2. Find two numbers that multiply to 6 and add to the coefficient of [tex]\(x\)[/tex] (which is -7):
We look for pairs of factors of 6: [tex]\( (1, 6), (2, 3) \)[/tex].
We need the pair that adds up to -7. The pair that works is [tex]\(-1\)[/tex] and [tex]\(-6\)[/tex] because:
[tex]\[-1 \times -6 = 6\][/tex]
[tex]\[-1 + -6 = -7.\][/tex]
3. Rewrite the middle term of the quadratic equation using these two numbers:
[tex]\[3x^2 - 7x + 2 = 3x^2 - 1x - 6x + 2.\][/tex]
4. Group the terms to factor by grouping:
[tex]\[ (3x^2 - 1x) + (-6x + 2).\][/tex]
5. Factor out the greatest common factor (GCF) from each group:
[tex]\[ x(3x - 1) - 2(3x - 1).\][/tex]
6. Notice that [tex]\(3x - 1\)[/tex] is a common factor in both groups:
[tex]\[ (x - 2)(3x - 1).\][/tex]
7. Set each factor equal to zero to solve for [tex]\(x\)[/tex]:
[tex]\[ x - 2 = 0 \quad \text{or} \quad 3x - 1 = 0.\][/tex]
8. Solve each equation:
[tex]\[ x - 2 = 0 \quad \Rightarrow \quad x = 2;\][/tex]
[tex]\[ 3x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{3}.\][/tex]
Thus, the solutions to the equation [tex]\(3x^2 - 7x + 2 = 0\)[/tex] are:
[tex]\[ x = 2 \quad \text{and} \quad x = \frac{1}{3}. \][/tex]
Written as:
[tex]\[ x_1 = 2.0 \][/tex]
[tex]\[ x_2 = 0.3333333333333333 \][/tex]
The solutions have been factored correctly and match our expectations.
