1. **State the problem:** Solve the quadratic equations by factoring. 2. **Recall the factoring method:** For a quadratic equation $ax^2 + bx + c = 0$, find two numbers that multiply to $ac$ and add to $b$. Then factor by grouping. 3. **Solve each equation:** **Equation 1: $9x^2 + 6x + 1 = 0$** - Multiply $a$ and $c$: $9 \times 1 = 9$ - Find two numbers that multiply to 9 and add to 6: 3 and 3 - Rewrite middle term: $9x^2 + 3x + 3x + 1 = 0$ - Factor by grouping: $3x(3x + 1) + 1(3x + 1) = 0$ - Factor out common binomial: $(3x + 1)(3x + 1) = 0$ - Set each factor to zero: $3x + 1 = 0$ - Solve for $x$: $x = -\frac{1}{3}$ - **Solution:** $x = -\frac{1}{3}$ (double root) **Equation 2: $2x^2 + 5x + 2 = 0$** - Multiply $a$ and $c$: $2 \times 2 = 4$ - Find two numbers that multiply to 4 and add to 5: 4 and 1 - Rewrite middle term: $2x^2 + 4x + x + 2 = 0$ - Factor by grouping: $2x(x + 2) + 1(x + 2) = 0$ - Factor out common binomial: $(x + 2)(2x + 1) = 0$ - Set each factor to zero: - $x + 2 = 0 \Rightarrow x = -2$ - $2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$ - **Solution:** $x = -2, -\frac{1}{2}$ **Equation 3: $x^2 - x - 12 = 0$** - Multiply $a$ and $c$: $1 \times (-12) = -12$ - Find two numbers that multiply to -12 and add to -1: -4 and 3 - Rewrite middle term: $x^2 - 4x + 3x - 12 = 0$ - Factor by grouping: $x(x - 4) + 3(x - 4) = 0$ - Factor out common binomial: $(x - 4)(x + 3) = 0$ - Set each factor to zero: - $x - 4 = 0 \Rightarrow x = 4$ - $x + 3 = 0 \Rightarrow x = -3$ - **Solution:** $x = 4, -3$ 4. **Match solutions to color codes:** - Equation 1 roots: $x = -\frac{1}{3}$ (double root) matches Black B and Black M - Equation 2 roots: $x = -2, -\frac{1}{2}$ matches Gray F and Gray L - Equation 3 roots: $x = 4, -3$ matches Black G 5. **Summary:** - Eq 1: $9x^2 + 6x + 1 = 0$, roots $x = -\frac{1}{3}$ (double root) - Eq 2: $2x^2 + 5x + 2 = 0$, roots $x = -2, -\frac{1}{2}$ - Eq 3: $x^2 - x - 12 = 0$, roots $x = 4, -3$