1. Problem: Find the vertex and y-intercept of the quadratic function $y = x^2 - 6x + 15$. 2. Use the vertex formula $h = -\frac{b}{2a}$ where $a=1$, $b=-6$. 3. Calculate $h = -\frac{-6}{2(1)} = \frac{6}{2} = 3$. 4. Substitute $x=3$ into the function to find $y$: $$y = (3)^2 - 6(3) + 15 = 9 - 18 + 15 = 6$$ 5. Vertex is $(3, 6)$. 6. Find y-intercept by substituting $x=0$: $$y = 0^2 - 6(0) + 15 = 15$$ 7. Y-intercept is $(0, 15)$. --- 1. Problem: Find the vertex and y-intercept of the quadratic function $y = 4x^2 - 15x + 9$. 2. Use $h = -\frac{b}{2a}$ with $a=4$, $b=-15$. 3. Calculate $h = -\frac{-15}{2(4)} = \frac{15}{8} = 1.875$. 4. Substitute $x=1.875$: $$y = 4(1.875)^2 - 15(1.875) + 9 = 4(3.515625) - 28.125 + 9 = 14.0625 - 28.125 + 9 = -5.0625$$ 5. Vertex is $(1.875, -5.0625)$. 6. Y-intercept at $x=0$: $$y = 9$$ 7. Y-intercept is $(0, 9)$. --- 1. Problem: Write an equation in standard form for the parabola passing through points $(1,5)$, $(3,7)$, $(6,25)$. 2. Assume $y = ax^2 + bx + c$. 3. Set up system: $$5 = a(1)^2 + b(1) + c = a + b + c$$ $$7 = a(3)^2 + b(3) + c = 9a + 3b + c$$ $$25 = a(6)^2 + b(6) + c = 36a + 6b + c$$ 4. Subtract first from second: $$7 - 5 = (9a - a) + (3b - b) + (c - c) \Rightarrow 2 = 8a + 2b$$ 5. Subtract first from third: $$25 - 5 = (36a - a) + (6b - b) + (c - c) \Rightarrow 20 = 35a + 5b$$ 6. Simplify: $$2 = 8a + 2b \Rightarrow 1 = 4a + b$$ $$20 = 35a + 5b \Rightarrow 4 = 7a + b$$ 7. Subtract equations: $$(4 = 7a + b) - (1 = 4a + b) \Rightarrow 3 = 3a \Rightarrow a = 1$$ 8. Substitute $a=1$ into $1 = 4a + b$: $$1 = 4(1) + b \Rightarrow b = 1 - 4 = -3$$ 9. Substitute $a=1$, $b=-3$ into $5 = a + b + c$: $$5 = 1 - 3 + c \Rightarrow c = 7$$ 10. Equation is: $$y = x^2 - 3x + 7$$ --- 1. Problem: Find vertex and maximum height of $y = -5x^2 + 40x + 60$ (golf ball path). 2. Use $h = -\frac{b}{2a}$ with $a = -5$, $b = 40$: $$h = -\frac{40}{2(-5)} = -\frac{40}{-10} = 4$$ 3. Substitute $x=4$: $$y = -5(4)^2 + 40(4) + 60 = -5(16) + 160 + 60 = -80 + 160 + 60 = 140$$ 4. Vertex is $(4, 140)$, maximum height is 140 meters. 5. To find horizontal distance when ball is 60 meters above hole, solve: $$-5x^2 + 40x + 60 = 60$$ 6. Simplify: $$-5x^2 + 40x = 0$$ 7. Factor: $$-5x(x - 8) = 0$$ 8. Solutions: $$x = 0 \text{ or } x = 8$$ 9. Horizontal distance traveled is $8$ meters. --- 1. Problem: Determine maximum profit for $y = -6x^2 + 36x + 50$. 2. Use $h = -\frac{b}{2a}$ with $a = -6$, $b = 36$: $$h = -\frac{36}{2(-6)} = -\frac{36}{-12} = 3$$ 3. Substitute $x=3$: $$y = -6(3)^2 + 36(3) + 50 = -6(9) + 108 + 50 = -54 + 108 + 50 = 104$$ 4. Maximum profit is 104 (thousands). --- 1. Problem: Solve $x^2 - 6x - 27 = 0$ by factoring. 2. Find factors of $-27$ that sum to $-6$: $-9$ and $3$. 3. Factor: $$(x - 9)(x + 3) = 0$$ 4. Solutions: $$x = 9 \text{ or } x = -3$$ --- 1. Problem: Solve $x^2 = 7x - 10$. 2. Rearrange: $$x^2 - 7x + 10 = 0$$ 3. Factor: $$(x - 5)(x - 2) = 0$$ 4. Solutions: $$x = 5 \text{ or } x = 2$$ --- 1. Problem: Solve $4x^2 + 4x = 3$. 2. Rearrange: $$4x^2 + 4x - 3 = 0$$ 3. Use quadratic formula: $$a=4, b=4, c=-3$$ 4. Calculate discriminant: $$b^2 - 4ac = 16 - 4(4)(-3) = 16 + 48 = 64$$ 5. Solutions: $$x = \frac{-4 \pm \sqrt{64}}{2(4)} = \frac{-4 \pm 8}{8}$$ 6. Two roots: $$x = \frac{-4 + 8}{8} = \frac{4}{8} = 0.5$$ $$x = \frac{-4 - 8}{8} = \frac{-12}{8} = -1.5$$ --- 1. Problem: Solve $5x^2 - 19x = -12$. 2. Rearrange: $$5x^2 - 19x + 12 = 0$$ 3. Use quadratic formula: $$a=5, b=-19, c=12$$ 4. Discriminant: $$(-19)^2 - 4(5)(12) = 361 - 240 = 121$$ 5. Solutions: $$x = \frac{19 \pm 11}{10}$$ 6. Roots: $$x = \frac{19 + 11}{10} = 3$$ $$x = \frac{19 - 11}{10} = 0.8$$ --- 1. Problem: Identify intervals where $y = x^2 - x - 30$ is positive. 2. Factor: $$(x - 6)(x + 5) = 0$$ 3. Roots at $x=6$ and $x=-5$. 4. Parabola opens upward ($a=1>0$), so $y>0$ outside roots. 5. Intervals: $$(-\infty, -5) \cup (6, \infty)$$ --- 1. Problem: Identify intervals where $y = x^2 + 11x + 28$ is positive. 2. Factor: $$(x + 7)(x + 4) = 0$$ 3. Roots at $x=-7$ and $x=-4$. 4. Parabola opens upward, so $y>0$ outside roots. 5. Intervals: $$(-\infty, -7) \cup (-4, \infty)$$ --- 1. Problem: For what values of $x$ is $(x + 6)^2 > 0$? 2. Square of any real number is $20$ except zero. 3. $(x + 6)^2 = 0$ when $x = -6$. 4. So $(x + 6)^2 > 0$ for all $x \neq -6$. --- 1. Problem: Find quadratic function for burrow with points $(2.5,0)$, $(8,0)$, $(5,-15)$. 2. Use factored form: $$y = a(x - 2.5)(x - 8)$$ 3. Substitute $(5, -15)$: $$-15 = a(5 - 2.5)(5 - 8) = a(2.5)(-3) = -7.5a$$ 4. Solve for $a$: $$a = \frac{-15}{-7.5} = 2$$ 5. Equation: $$y = 2(x - 2.5)(x - 8)$$ 6. Expand: $$y = 2(x^2 - 10.5x + 20) = 2x^2 - 21x + 40$$ --- 1. Problem: Write product $(5 - 3i)(2 + i)$ in form $a + bi$. 2. Multiply: $$5(2) + 5(i) - 3i(2) - 3i(i) = 10 + 5i - 6i - 3i^2$$ 3. Simplify: $$10 - i - 3(-1) = 10 - i + 3 = 13 - i$$ --- 1. Problem: Divide $\frac{2 - 3i}{1 + 2i}$ and write in $a + bi$. 2. Multiply numerator and denominator by conjugate $1 - 2i$: $$\frac{(2 - 3i)(1 - 2i)}{(1 + 2i)(1 - 2i)}$$ 3. Denominator: $$1 - (2i)^2 = 1 - (-4) = 5$$ 4. Numerator: $$2(1) - 2(2i) - 3i(1) + 3i(2i) = 2 - 4i - 3i + 6i^2 = 2 - 7i + 6(-1) = 2 - 7i - 6 = -4 - 7i$$ 5. Result: $$\frac{-4 - 7i}{5} = -\frac{4}{5} - \frac{7}{5}i$$ --- 1. Problem: Correct error in multiplying $(2 - 3i)(4 + i)$. 2. Correct multiplication: $$2(4) + 2(i) - 3i(4) - 3i(i) = 8 + 2i - 12i - 3i^2$$ 3. Simplify: $$8 - 10i - 3(-1) = 8 - 10i + 3 = 11 - 10i$$ 4. Error was missing sign change for $-3i(i)$ term. --- 1. Problem: Find current $I$ given $E = 35 + 10i$, $Z = 4 + 4i$, $E = IZ$. 2. Solve for $I$: $$I = \frac{E}{Z} = \frac{35 + 10i}{4 + 4i}$$ 3. Multiply numerator and denominator by conjugate $4 - 4i$: $$I = \frac{(35 + 10i)(4 - 4i)}{(4 + 4i)(4 - 4i)}$$ 4. Denominator: $$16 - (4i)^2 = 16 - (-16) = 32$$ 5. Numerator: $$35(4) - 35(4i) + 10i(4) - 10i(4i) = 140 - 140i + 40i - 40i^2 = 140 - 100i + 40 = 180 - 100i$$ 6. Result: $$I = \frac{180 - 100i}{32} = \frac{180}{32} - \frac{100}{32}i = 5.625 - 3.125i$$ --- 1. Problem: Solve $0 = x^2 - 16x + 36$ by completing the square. 2. Rearrange: $$x^2 - 16x = -36$$ 3. Half of $-16$ is $-8$, square is $64$. 4. Add $64$ both sides: $$x^2 - 16x + 64 = -36 + 64$$ 5. Left side is perfect square: $$(x - 8)^2 = 28$$ --- 1. Problem: Solve $0 = 4x^2 - 28x - 42$ by completing the square. 2. Divide by 4: $$x^2 - 7x = \frac{42}{4} = 10.5$$ 3. Half of $-7$ is $-3.5$, square is $12.25$. 4. Add $12.25$ both sides: $$x^2 - 7x + 12.25 = 10.5 + 12.25 = 22.75$$ 5. Left side: $$(x - 3.5)^2 = 22.75$$ --- 1. Problem: Solve $x^2 - 24x - 82 = 0$ by completing the square. 2. Rearrange: $$x^2 - 24x = 82$$ 3. Half of $-24$ is $-12$, square is $144$. 4. Add $144$ both sides: $$x^2 - 24x + 144 = 82 + 144 = 226$$ 5. Left side: $$(x - 12)^2 = 226$$ 6. Solutions: $$x = 12 \pm \sqrt{226}$$ --- 1. Problem: Solve $-3x^2 - 42x = 18$ by completing the square. 2. Divide by $-3$: $$x^2 + 14x = -6$$ 3. Half of $14$ is $7$, square is $49$. 4. Add $49$ both sides: $$x^2 + 14x + 49 = -6 + 49 = 43$$ 5. Left side: $$(x + 7)^2 = 43$$ 6. Solutions: $$x = -7 \pm \sqrt{43}$$ --- 1. Problem: Solve $4x^2 = 16x + 25$ by completing the square. 2. Rearrange: $$4x^2 - 16x = 25$$ 3. Divide by 4: $$x^2 - 4x = \frac{25}{4}$$ 4. Half of $-4$ is $-2$, square is $4$. 5. Add $4$ both sides: $$x^2 - 4x + 4 = \frac{25}{4} + 4 = \frac{25}{4} + \frac{16}{4} = \frac{41}{4}$$ 6. Left side: $$(x - 2)^2 = \frac{41}{4}$$ 7. Solutions: $$x = 2 \pm \frac{\sqrt{41}}{2}$$ --- 1. Problem: Solve $12 + x^2 = 15x$ by completing the square. 2. Rearrange: $$x^2 - 15x = -12$$ 3. Half of $-15$ is $-7.5$, square is $56.25$. 4. Add $56.25$ both sides: $$x^2 - 15x + 56.25 = -12 + 56.25 = 44.25$$ 5. Left side: $$(x - 7.5)^2 = 44.25$$ 6. Solutions: $$x = 7.5 \pm \sqrt{44.25}$$ --- 1. Problem: When does ball hit ground for $f(x) = -4.9x^2 + 24.5x + 1$ with roots approx $-0.04$ and $5.04$? 2. Negative root is non-physical (time cannot be negative). 3. Ball hits ground at $x \approx 5.04$ seconds. --- 1. Problem: Find prices where $P(x) = -100x^2 + 46000x - 2100000 = 0$. 2. Use quadratic formula: $$a = -100, b = 46000, c = -2100000$$ 3. Discriminant: $$b^2 - 4ac = 46000^2 - 4(-100)(-2100000) = 2.116 \times 10^9 - 8.4 \times 10^8 = 1.276 \times 10^9$$ 4. Roots: $$x = \frac{-46000 \pm \sqrt{1.276 \times 10^9}}{2(-100)}$$ 5. Calculate roots: $$x = \frac{-46000 \pm 35720}{-200}$$ 6. First root: $$x = \frac{-46000 + 35720}{-200} = \frac{-10280}{-200} = 51.4$$ 7. Second root: $$x = \frac{-46000 - 35720}{-200} = \frac{-81720}{-200} = 408.6$$ 8. Prices for zero profit are approximately 51.4 and 408.6. --- 1. Problem: Number of real roots of $3x^2 - 8x + 1 = 0$. 2. Discriminant: $$(-8)^2 - 4(3)(1) = 64 - 12 = 52 > 0$$ 3. Two real roots. --- 1. Problem: Solve $x^2 - 16x + 24 = 0$ using quadratic formula. 2. Discriminant: $$256 - 96 = 160$$ 3. Roots: $$x = \frac{16 \pm \sqrt{160}}{2} = 8 \pm 2\sqrt{10}$$ --- 1. Problem: Solve $x^2 + 5x + 2 = 0$. 2. Discriminant: $$25 - 8 = 17$$ 3. Roots: $$x = \frac{-5 \pm \sqrt{17}}{2}$$ --- 1. Problem: Solve $2x^2 - 18x + 5 = 0$. 2. Discriminant: $$324 - 40 = 284$$ 3. Roots: $$x = \frac{18 \pm \sqrt{284}}{4}$$ --- 1. Problem: Solve $3x^2 - 5x - 19 = 0$. 2. Discriminant: $$25 + 228 = 253$$ 3. Roots: $$x = \frac{5 \pm \sqrt{253}}{6}$$ --- 1. Problem: Number and type of solutions for $x^2 - 24x + 19 = 0$. 2. Discriminant: $$576 - 76 = 500 > 0$$ 3. Two real roots. --- 1. Problem: Number and type of solutions for $3x^2 - 8x + 12 = 0$. 2. Discriminant: $$64 - 144 = -80 < 0$$ 3. Two non-real roots. --- 1. Problem: Find $k$ for $4x^2 - kx + 4 = 0$ to have one real solution. 2. Discriminant zero: $$k^2 - 4(4)(4) = 0 \Rightarrow k^2 = 64 \Rightarrow k = \pm 8$$ --- 1. Problem: Why does $f(x) = x^2 + 4x + 5$ cross y-axis but not x-axis? 2. Discriminant: $$16 - 20 = -4 < 0$$ 3. No real roots, so no x-intercepts. 4. Y-intercept at $x=0$ is $5$, so graph crosses y-axis. --- 1. Problem: Find speeds where $C(x) = 0.0045x^2 - 0.47x + 139 > 130$. 2. Solve inequality: $$0.0045x^2 - 0.47x + 139 > 130$$ 3. Simplify: $$0.0045x^2 - 0.47x + 9 > 0$$ 4. Solve equality: $$0.0045x^2 - 0.47x + 9 = 0$$ 5. Discriminant: $$(-0.47)^2 - 4(0.0045)(9) = 0.2209 - 0.162 = 0.0589$$ 6. Roots: $$x = \frac{0.47 \pm \sqrt{0.0589}}{2(0.0045)} = \frac{0.47 \pm 0.2427}{0.009}$$ 7. Calculate roots: $$x_1 = \frac{0.47 - 0.2427}{0.009} = 25.86$$ $$x_2 = \frac{0.47 + 0.2427}{0.009} = 79.19$$ 8. Parabola opens upward, so $C(x) > 130$ outside roots. 9. Speeds: $$x < 25.86 \text{ or } x > 79.19$$