Certainly! Let's go through the complete solution step-by-step.
We need to factor the quadratic equation [tex]\(2x^2 + 7x + 6 = 0\)[/tex].
1. First, identify the quadratic expression: [tex]\(2x^2 + 7x + 6\)[/tex].
2. To factor it, we look for two numbers that multiply to the product of the coefficient of [tex]\(x^2\)[/tex] (which is 2) and the constant term (which is 6). These numbers must also add up to the coefficient of [tex]\(x\)[/tex] (which is 7).
- The product of 2 (coefficient of [tex]\(x^2\)[/tex]) and 6 (constant term) is [tex]\(2 \times 6 = 12\)[/tex].
- We need two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4.
3. Rewrite the middle term using these two numbers:
[tex]\[
2x^2 + 7x + 6 = 2x^2 + 3x + 4x + 6
\][/tex]
4. Group the terms:
[tex]\[
(2x^2 + 3x) + (4x + 6)
\][/tex]
5. Factor by grouping:
[tex]\[
x(2x + 3) + 2(2x + 3)
\][/tex]
6. Notice that [tex]\(2x + 3\)[/tex] is a common factor:
[tex]\[
(x + 2)(2x + 3)
\][/tex]
So, we have factored [tex]\(2x^2 + 7x + 6\)[/tex] as [tex]\((x + 2)(2x + 3)\)[/tex].
Now, solve the factored equation [tex]\((x + 2)(2x + 3) = 0\)[/tex] using the zero product property:
7. Set each factor to zero:
[tex]\[
x + 2 = 0 \quad \text{or} \quad 2x + 3 = 0
\][/tex]
8. Solve each equation for [tex]\(x\)[/tex]:
For [tex]\(x + 2 = 0\)[/tex]:
[tex]\[
x = -2
\][/tex]
For [tex]\(2x + 3 = 0\)[/tex]:
[tex]\[
2x = -3 \implies x = -\frac{3}{2} \implies x = -1.5
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(2x^2 + 7x + 6 = 0\)[/tex] are:
[tex]\[
x = -2 \quad \text{and} \quad x = -1.5
\][/tex]
These are the roots of the quadratic equation.
