$$ y^{2}-y-72=0 \quad $$ 20. $$ \quad x^{2}-12 x=-11 $$

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UpStudy AI · Accepted Answer

Sure, let's solve both quadratic equations step by step.

---

### **1. Solving $$ y^{2} - y - 72 = 0 $$**

This is a quadratic equation in the standard form $$ ay^{2} + by + c = 0 $$, where:
- $$ a = 1 $$
- $$ b = -1 $$
- $$ c = -72 $$

**Step 1: Identify the coefficients.**
$$
a = 1,\quad b = -1,\quad c = -72
$$

**Step 2: Apply the Quadratic Formula.**
The quadratic formula is:
$$
y = \frac{ -b \pm \sqrt{b^{2} - 4ac} }{2a}
$$
Plugging in the values:
$$
y = \frac{ -(-1) \pm \sqrt{(-1)^{2} - 4 \times 1 \times (-72)} }{2 \times 1}
$$
$$
y = \frac{ 1 \pm \sqrt{1 + 288} }{2}
$$
$$
y = \frac{ 1 \pm \sqrt{289} }{2}
$$

**Step 3: Compute the square root.**
$$
\sqrt{289} = 17
$$

**Step 4: Find the two possible solutions.**
$$
y = \frac{1 + 17}{2} = \frac{18}{2} = 9
$$
$$
y = \frac{1 - 17}{2} = \frac{-16}{2} = -8
$$

****_Solutions for $$ y $$:_**
$$
y = 9 \quad \text{and} \quad y = -8
$$

---

### **2. Solving $$ x^{2} - 12x = -11 $$**

First, let's write the equation in standard quadratic form:
$$
x^{2} - 12x + 11 = 0
$$

**Step 1: Identify the coefficients.**
$$
a = 1,\quad b = -12,\quad c = 11
$$

**Step 2: Apply the Quadratic Formula.**
$$
x = \frac{ -b \pm \sqrt{b^{2} - 4ac} }{2a}
$$
Plugging in the values:
$$
x = \frac{ -(-12) \pm \sqrt{(-12)^{2} - 4 \times 1 \times 11} }{2 \times 1}
$$
$$
x = \frac{ 12 \pm \sqrt{144 - 44} }{2}
$$
$$
x = \frac{ 12 \pm \sqrt{100} }{2}
$$

**Step 3: Compute the square root.**
$$
\sqrt{100} = 10
$$

**Step 4: Find the two possible solutions.**
$$
x = \frac{12 + 10}{2} = \frac{22}{2} = 11
$$
$$
x = \frac{12 - 10}{2} = \frac{2}{2} = 1
$$

**_Solutions for $$ x $$:_**
$$
x = 11 \quad \text{and} \quad x = 1
$$

---

**Summary:**
- For the equation $$ y^{2} - y - 72 = 0 $$, the solutions are **$$ y = 9 $$** and **$$ y = -8 $$**.
- For the equation $$ x^{2} - 12x = -11 $$, the solutions are **$$ x = 11 $$** and **$$ x = 1 $$**.
The solutions are: - For $$ y^{2} - y - 72 = 0 $$: $$ y = 9 $$ and $$ y = -8 $$ - For $$ x^{2} - 12x = -11 $$: $$ x = 11 $$ and $$ x = 1 $$

\( y^{2}-y-72=0 \quad \) 20. \( \quad x^{2}-12 x=-11 \)

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