find the number of x-intercepts.
3. $y = x^2 + 6x - 7$
4. $y = -x^2 + 6x - 9$
solve using quadratic formula. use the discriminant to determine the number and types of solutions.
5. $3n^2 - 5n - 8 = 0$
6. $x^2 + 10x = -21$
7. $10x^2 + 6 = 9x$
8. $8x^2 - 3 = 0$
9. $12n^2 - 10n - 9 = 2n - 8$
10. $6p^2 = 8p - 6$

Question

Accepted Answer

### Problem 3: $ y = x^2 + 6x - 7 $ # Explanation: ## Step 1: Recall x-intercept definition To find the number of $ x $-intercepts, we set $ y = 0 $ and solve the quadratic equation $ x^2 + 6x - 7 = 0 $. We can use the discriminant $ D = b^2 - 4ac $ for a quadratic $ ax^2 + bx + c = 0 $. Here, $ a = 1 $, $ b = 6 $, $ c = -7 $. ## Step 2: Calculate the discriminant $$ D = 6^2 - 4(1)(-7) = 36 + 28 = 64 $$ Since $ D = 64 > 0 $, the quadratic equation has two distinct real roots. So the number of $ x $-intercepts is 2. ### Problem 4: $ y = -x^2 + 6x - 9 $ # Explanation: ## Step 1: Set $ y = 0 $ We solve $ -x^2 + 6x - 9 = 0 $, or multiply by -1: $ x^2 - 6x + 9 = 0 $. For $ ax^2 + bx + c = 0 $, here $ a = 1 $, $ b = -6 $, $ c = 9 $. ## Step 2: Calculate the discriminant $$ D = (-6)^2 - 4(1)(9) = 36 - 36 = 0 $$ Since $ D = 0 $, the quadratic equation has one real root (a repeated root). So the number of $ x $-intercepts is 1. ### Problem 5: $ 3n^2 - 5n - 8 = 0 $ # Explanation: ## Step 1: Identify $ a, b, c $ For $ 3n^2 - 5n - 8 = 0 $, $ a = 3 $, $ b = -5 $, $ c = -8 $. ## Step 2: Quadratic Formula $ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ First, calculate discriminant $ D = (-5)^2 - 4(3)(-8) = 25 + 96 = 121 $ Then, $$ n = \frac{-(-5) \pm \sqrt{121}}{2(3)} = \frac{5 \pm 11}{6} $$ ## Step 3: Find the two solutions For $ + $: $ \frac{5 + 11}{6} = \frac{16}{6} = \frac{8}{3} $ For $ - $: $ \frac{5 - 11}{6} = \frac{-6}{6} = -1 $ So solutions are $ n = \frac{8}{3} $ and $ n = -1 $. Discriminant $ D = 121 > 0 $, so two distinct real solutions. ### Problem 6: $ x^2 + 10x = -21 $ (rewrite as $ x^2 + 10x + 21 = 0 $) # Explanation: ## Step 1: Identify $ a, b, c $ Here, $ a = 1 $, $ b = 10 $, $ c = 21 $. ## Step 2: Calculate discriminant $$ D = 10^2 - 4(1)(21) = 100 - 84 = 16 $$ ## Step 3: Quadratic Formula $ x = \frac{-b \pm \sqrt{D}}{2a} $ $$ x = \frac{-10 \pm \sqrt{16}}{2(1)} = \frac{-10 \pm 4}{2} $$ ## Step 4: Find solutions For $ + $: $ \frac{-10 + 4}{2} = \frac{-6}{2} = -3 $ For $ - $: $ \frac{-10 - 4}{2} = \frac{-14}{2} = -7 $ Solutions: $ x = -3 $, $ x = -7 $. Discriminant $ D = 16 > 0 $, two distinct real solutions. ### Problem 7: $ 10x^2 + 6 = 9x $ (rewrite as $ 10x^2 - 9x + 6 = 0 $) # Explanation: ## Step 1: Identify $ a, b, c $ $ a = 10 $, $ b = -9 $, $ c = 6 $ ## Step 2: Calculate discriminant $$ D = (-9)^2 - 4(10)(6) = 81 - 240 = -159 $$ Since $ D < 0 $, there are no real solutions (two complex solutions). Using Quadratic Formula: $$ x = \frac{-(-9) \pm \sqrt{-159}}{2(10)} = \frac{9 \pm i\sqrt{159}}{20} $$ ### Problem 8: $ 8x^2 - 3 = 0 $ # Explanation: ## Step 1: Identify $ a, b, c $ $ a = 8 $, $ b = 0 $, $ c = -3 $ ## Step 2: Calculate discriminant $$ D = 0^2 - 4(8)(-3) = 0 + 96 = 96 $$ ## Step 3: Quadratic Formula $ x = \frac{-b \pm \sqrt{D}}{2a} $ $$ x = \frac{-0 \pm \sqrt{96}}{2(8)} = \frac{\pm 4\sqrt{6}}{16} = \frac{\pm \sqrt{6}}{4} $$ Discriminant $ D = 96 > 0 $, two distinct real solutions. ### Problem 9: $ 12n^2 - 10n - 9 = 2n - 8 $ (rewrite as $ 12n^2 - 12n - 1 = 0 $) # Explanation: ## Step 1: Identify $ a, b, c $ $ a = 12 $, $ b = -12 $, $ c = -1 $ ## Step 2: Calculate discriminant $$ D = (-12)^2 - 4(12)(-1) = 144 + 48 = 192 $$ ## Step 3: Quadratic Formula $ n = \frac{-b \pm \sqrt{D}}{2a} $ $$ n = \frac{-(-12) \pm \sqrt{192}}{2(12)} = \frac{12 \pm 8\sqrt{3}}{24} = \frac{3 \pm 2\sqrt{3}}{6} $$ Discriminant $ D = 192 > 0 $, two distinct real solutions. ### Problem 10: $ 6p^2 = 8p - 6 $ (rewrite as $ 6p^2 - 8p + 6 = 0 $, simplify by dividing by 2: $ 3p^2 - 4p + 3 = 0 $) # Explanation: ## Step 1: Identify $ a, b, c $ $ a = 3 $, $ b = -4 $, $ c = 3 $ ## Step 2: Calculate discriminant $$ D = (-4)^2 - 4(3)(3) = 16 - 36 = -20 $$ Since $ D < 0 $, there are no real solutions (two complex solutions). Using Quadratic Formula: $$ p = \frac{-(-4) \pm \sqrt{-20}}{2(3)} = \frac{4 \pm 2i\sqrt{5}}{6} = \frac{2 \pm i\sqrt{5}}{3} $$ ### Answers: 3. Number of $ x $-intercepts: $ \boldsymbol{2} $ 4. Number of $ x $-intercepts: $ \boldsymbol{1} $ 5. Solutions: $ \boldsymbol{n = \frac{8}{3}, n = -1} $ (two real solutions) 6. Solutions: $ \boldsymbol{x = -3, x = -7} $ (two real solutions) 7. Solutions: $ \boldsymbol{\text{No real solutions (complex: } \frac{9 \pm i\sqrt{159}}{20}\text{)}} $ 8. Solutions: $ \boldsymbol{x = \frac{\sqrt{6}}{4}, x = -\frac{\sqrt{6}}{4}} $ (two real solutions) 9. Solutions: $ \boldsymbol{n = \frac{3 + 2\sqrt{3}}{6}, n = \frac{3 - 2\sqrt{3}}{6}} $ (two real solutions) 10. Solutions: $ \boldsymbol{\text{No real solutions (complex: } \frac{2 \pm i\sqrt{5}}{3}\text{)}} $

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QUESTION IMAGE

Question

Explanation:

Problem 3: \( y = x^2 + 6x - 7 \)

Step 1: Recall x-intercept definition

Step 2: Calculate the discriminant

Problem 4: \( y = -x^2 + 6x - 9 \)

Step 1: Set \( y = 0 \)

Step 2: Calculate the discriminant

Problem 5: \( 3n^2 - 5n - 8 = 0 \)

Step 1: Identify \( a, b, c \)

Step 2: Quadratic Formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step 3: Find the two solutions

Problem 6: \( x^2 + 10x = -21 \) (rewrite as \( x^2 + 10x + 21 = 0 \))

Answer: