topic 5: solve by using the quadratic formula
1) $p^2 + 14p - 38 = 0$
2) $v^2 + 6v - 59 = 0$
3) $a^2 + 14a - 51 = 0$
4) $x^2 - 12x + 11 = 0$
5) $x^2 + 6x + 8 = 0$
6) $n^2 - 2n - 3 = 0$
quadratic formula:
for $ax^2 + bx + c = 0$,
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
mathbits.com

Question

Accepted Answer

### Problem 1: $ p^2 + 14p - 38 = 0 $
# Explanation:
## Step 1: Identify $ a, b, c $
For $ p^2 + 14p - 38 = 0 $, $ a = 1 $, $ b = 14 $, $ c = -38 $.
## Step 2: Apply Quadratic Formula
Quadratic formula: $ p=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $
Substitute $ a = 1 $, $ b = 14 $, $ c = -38 $:
$ p=\frac{-14\pm\sqrt{14^2 - 4(1)(-38)}}{2(1)} $
## Step 3: Simplify the discriminant
Calculate $ 14^2 - 4(1)(-38) = 196 + 152 = 348 $
So, $ p=\frac{-14\pm\sqrt{348}}{2} $
Simplify $ \sqrt{348}=\sqrt{4\times87}=2\sqrt{87} $
Then $ p=\frac{-14\pm2\sqrt{87}}{2}=-7\pm\sqrt{87} $
# Answer:
$ p = -7 + \sqrt{87} $ or $ p = -7 - \sqrt{87} $ (or approximately $ p \approx 2.327 $ or $ p \approx -16.327 $)

### Problem 2: $ v^2 + 6v - 59 = 0 $
# Explanation:
## Step 1: Identify $ a, b, c $
For $ v^2 + 6v - 59 = 0 $, $ a = 1 $, $ b = 6 $, $ c = -59 $.
## Step 2: Apply Quadratic Formula
$ v=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $
Substitute $ a = 1 $, $ b = 6 $, $ c = -59 $:
$ v=\frac{-6\pm\sqrt{6^2 - 4(1)(-59)}}{2(1)} $
## Step 3: Simplify the discriminant
Calculate $ 6^2 - 4(1)(-59) = 36 + 236 = 272 $
So, $ v=\frac{-6\pm\sqrt{272}}{2} $
Simplify $ \sqrt{272}=\sqrt{16\times17}=4\sqrt{17} $
Then $ v=\frac{-6\pm4\sqrt{17}}{2}=-3\pm2\sqrt{17} $
# Answer:
$ v = -3 + 2\sqrt{17} $ or $ v = -3 - 2\sqrt{17} $ (or approximately $ v \approx 5.246 $ or $ v \approx -11.246 $)

### Problem 3: $ a^2 + 14a - 51 = 0 $
# Explanation:
## Step 1: Identify $ a, b, c $
For $ a^2 + 14a - 51 = 0 $, $ a = 1 $, $ b = 14 $, $ c = -51 $.
## Step 2: Apply Quadratic Formula
$ a=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $
Substitute $ a = 1 $, $ b = 14 $, $ c = -51 $:
$ a=\frac{-14\pm\sqrt{14^2 - 4(1)(-51)}}{2(1)} $
## Step 3: Simplify the discriminant
Calculate $ 14^2 - 4(1)(-51) = 196 + 204 = 400 $
So, $ a=\frac{-14\pm\sqrt{400}}{2}=\frac{-14\pm20}{2} $
## Step 4: Solve for $ a $
Case 1: $ \frac{-14 + 20}{2}=\frac{6}{2}=3 $
Case 2: $ \frac{-14 - 20}{2}=\frac{-34}{2}=-17 $
# Answer:
$ a = 3 $ or $ a = -17 $

### Problem 4: $ x^2 - 12x + 11 = 0 $
# Explanation:
## Step 1: Identify $ a, b, c $
For $ x^2 - 12x + 11 = 0 $, $ a = 1 $, $ b = -12 $, $ c = 11 $.
## Step 2: Apply Quadratic Formula
$ x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $
Substitute $ a = 1 $, $ b = -12 $, $ c = 11 $:
$ x=\frac{-(-12)\pm\sqrt{(-12)^2 - 4(1)(11)}}{2(1)}=\frac{12\pm\sqrt{144 - 44}}{2} $
## Step 3: Simplify the discriminant
Calculate $ 144 - 44 = 100 $
So, $ x=\frac{12\pm\sqrt{100}}{2}=\frac{12\pm10}{2} $
## Step 4: Solve for $ x $
Case 1: $ \frac{12 + 10}{2}=\frac{22}{2}=11 $
Case 2: $ \frac{12 - 10}{2}=\frac{2}{2}=1 $
# Answer:
$ x = 11 $ or $ x = 1 $

### Problem 5: $ x^2 + 6x + 8 = 0 $
# Explanation:
## Step 1: Identify $ a, b, c $
For $ x^2 + 6x + 8 = 0 $, $ a = 1 $, $ b = 6 $, $ c = 8 $.
## Step 2: Apply Quadratic Formula
$ x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $
Substitute $ a = 1 $, $ b = 6 $, $ c = 8 $:
$ x=\frac{-6\pm\sqrt{6^2 - 4(1)(8)}}{2(1)}=\frac{-6\pm\sqrt{36 - 32}}{2} $
## Step 3: Simplify the discriminant
Calculate $ 36 - 32 = 4 $
So, $ x=\frac{-6\pm\sqrt{4}}{2}=\frac{-6\pm2}{2} $
## Step 4: Solve for $ x $
Case 1: $ \frac{-6 + 2}{2}=\frac{-4}{2}=-2 $
Case 2: $ \frac{-6 - 2}{2}=\frac{-8}{2}=-4 $
# Answer:
$ x = -2 $ or $ x = -4 $

### Problem 6: $ n^2 - 2n - 3 = 0 $
# Explanation:
## Step 1: Identify $ a, b, c $
For $ n^2 - 2n - 3 = 0 $, $ a = 1 $, $ b = -2 $, $ c = -3 $.
## Step 2: Apply Quadratic Formula
$ n=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $
Substitute $ a = 1 $, $ b = -2 $, $ c = -3 $:
$ n=\frac{-(-2)\pm\sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}=\frac{2\pm\sqrt{4 + 12}}{2} $
## Step 3: Simplify the discriminant
Calculate $ 4 + 12 = 16 $
So, $ n=\frac{2\pm\sqrt{16}}{2}=\frac{2\pm4}{2} $
## Step 4: Solve for $ n $
Case 1: $ \frac{2 + 4}{2}=\frac{6}{2}=3 $
Case 2: $ \frac{2 - 4}{2}=\frac{-2}{2}=-1 $
# Answer:
$ n = 3 $ or $ n = -1 $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( p^2 + 14p - 38 = 0 \)

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

Step 2: Apply Quadratic Formula

Step 3: Simplify the discriminant

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

Step 2: Apply Quadratic Formula

Step 3: Simplify the discriminant

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

Step 2: Apply Quadratic Formula

Step 3: Simplify the discriminant

Step 4: Solve for \( a \)

Answer:

Problem 2: \( v^2 + 6v - 59 = 0 \)