1. **Problem Statement:** Factor the quadratic equations and analyze the given quadratic graph. 2. **Factoring Quadratics:** The general form is $ax^2 + bx + c$. To factor, find two numbers that multiply to $ac$ and add to $b$. 3. **a) Factor $x^2 - 3x - 10$:** - Here, $a=1$, $b=-3$, $c=-10$. - Find two numbers that multiply to $1 \times (-10) = -10$ and add to $-3$. - These numbers are $-5$ and $2$. - Rewrite: $x^2 - 5x + 2x - 10$. - Group: $(x^2 - 5x) + (2x - 10)$. - Factor each group: $x(x - 5) + 2(x - 5)$. - Factor out common binomial: $(x - 5)(x + 2)$. 4. **b) Factor $4x^2 - 24x + 20$:** - $a=4$, $b=-24$, $c=20$. - First, factor out the greatest common factor (GCF): $4(x^2 - 6x + 5)$. - Now factor inside parentheses: find two numbers that multiply to $1 \times 5 = 5$ and add to $-6$. - These are $-5$ and $-1$. - Rewrite: $x^2 - 5x - x + 5$. - Group: $(x^2 - 5x) - (x - 5)$. - Factor each group: $x(x - 5) - 1(x - 5)$. - Factor out common binomial: $(x - 5)(x - 1)$. - So full factorization: $4(x - 5)(x - 1)$. 5. **c) Factor $3x^2 + 5x + 2$:** - $a=3$, $b=5$, $c=2$. - Find two numbers that multiply to $3 \times 2 = 6$ and add to $5$. - These are $2$ and $3$. - Rewrite: $3x^2 + 2x + 3x + 2$. - Group: $(3x^2 + 2x) + (3x + 2)$. - Factor each group: $x(3x + 2) + 1(3x + 2)$. - Factor out common binomial: $(3x + 2)(x + 1)$. 6. **Graph Analysis:** - Zeros (x-intercepts) are where the graph crosses the x-axis: approximately $x = -5$ and $x = 1$. - Axis of symmetry formula: $x = \frac{x_1 + x_2}{2} = \frac{-5 + 1}{2} = -2$. - Optimal value is the vertex's y-coordinate: approximately $-8$. - Vertex coordinates: $(-2, -8)$. - Equation in factored form uses zeros: $y = a(x + 5)(x - 1)$. - To find $a$, use vertex or another point. Using vertex $(-2, -8)$: $$-8 = a(-2 + 5)(-2 - 1) = a(3)(-3) = -9a$$ $$a = \frac{-8}{-9} = \frac{8}{9}$$ - Final equation: $$y = \frac{8}{9}(x + 5)(x - 1)$$ **Final answers:** - a) $(x - 5)(x + 2)$ - b) $4(x - 5)(x - 1)$ - c) $(3x + 2)(x + 1)$ - Graph equation: $y = \frac{8}{9}(x + 5)(x - 1)$ - Zeros: $-5$, $1$ - Axis of symmetry: $x = -2$ - Vertex: $(-2, -8)$ - Optimal value: $-8$