14. solve the equation for all real solutions in simplest form.
$5v^2 - 3v + 1 = 4v$
15. solve the equation for all real solutions in simplest form.
$5b^2 + 3b - 4 = -1$
19. solve the equation for all real solutions in simplest form.
$10b^2 - 12b + 3 = 5b^2$
20. solve the equation for all real solutions in simplest form.
$-4p^2 + 8p + 3 = -5p^2$
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Question

Accepted Answer

Let's solve question 14: $ 5v^2 - 3v + 1 = 4v $

# Explanation:
## Step 1: Rearrange the equation to standard quadratic form $ ax^2 + bx + c = 0 $
Subtract $ 4v $ from both sides:
$ 5v^2 - 3v - 4v + 1 = 0 $
Simplify the like terms:
$ 5v^2 - 7v + 1 = 0 $

## Step 2: Use the quadratic formula $ v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
Here, $ a = 5 $, $ b = -7 $, $ c = 1 $
First, calculate the discriminant $ D = b^2 - 4ac $
$ D = (-7)^2 - 4(5)(1) = 49 - 20 = 29 $

## Step 3: Substitute into the quadratic formula
$ v = \frac{-(-7) \pm \sqrt{29}}{2(5)} = \frac{7 \pm \sqrt{29}}{10} $

# Answer:
The solutions are $ v = \frac{7 + \sqrt{29}}{10} $ and $ v = \frac{7 - \sqrt{29}}{10} $

Let's solve question 15: $ 5b^2 + 3b - 4 = -1 $

# Explanation:
## Step 1: Rearrange the equation to standard quadratic form $ ax^2 + bx + c = 0 $
Add $ 1 $ to both sides:
$ 5b^2 + 3b - 4 + 1 = 0 $
Simplify the like terms:
$ 5b^2 + 3b - 3 = 0 $

## Step 2: Use the quadratic formula $ b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
Here, $ a = 5 $, $ b = 3 $, $ c = -3 $
First, calculate the discriminant $ D = b^2 - 4ac $
$ D = (3)^2 - 4(5)(-3) = 9 + 60 = 69 $

## Step 3: Substitute into the quadratic formula
$ b = \frac{-3 \pm \sqrt{69}}{2(5)} = \frac{-3 \pm \sqrt{69}}{10} $

# Answer:
The solutions are $ b = \frac{-3 + \sqrt{69}}{10} $ and $ b = \frac{-3 - \sqrt{69}}{10} $

Let's solve question 19: $ 10b^2 - 12b + 3 = 5b^2 $

# Explanation:
## Step 1: Rearrange the equation to standard quadratic form $ ax^2 + bx + c = 0 $
Subtract $ 5b^2 $ from both sides:
$ 10b^2 - 5b^2 - 12b + 3 = 0 $
Simplify the like terms:
$ 5b^2 - 12b + 3 = 0 $

## Step 2: Use the quadratic formula $ b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
Here, $ a = 5 $, $ b = -12 $, $ c = 3 $
First, calculate the discriminant $ D = b^2 - 4ac $
$ D = (-12)^2 - 4(5)(3) = 144 - 60 = 84 $
Simplify $ \sqrt{84} = 2\sqrt{21} $

## Step 3: Substitute into the quadratic formula
$ b = \frac{-(-12) \pm 2\sqrt{21}}{2(5)} = \frac{12 \pm 2\sqrt{21}}{10} = \frac{6 \pm \sqrt{21}}{5} $ (divided numerator and denominator by 2)

# Answer:
The solutions are $ b = \frac{6 + \sqrt{21}}{5} $ and $ b = \frac{6 - \sqrt{21}}{5} $

Let's solve question 20: $ -4p^2 + 8p + 3 = -5p^2 $

# Explanation:
## Step 1: Rearrange the equation to standard quadratic form $ ax^2 + bx + c = 0 $
Add $ 5p^2 $ to both sides:
$ -4p^2 + 5p^2 + 8p + 3 = 0 $
Simplify the like terms:
$ p^2 + 8p + 3 = 0 $

## Step 2: Use the quadratic formula $ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
Here, $ a = 1 $, $ b = 8 $, $ c = 3 $
First, calculate the discriminant $ D = b^2 - 4ac $
$ D = (8)^2 - 4(1)(3) = 64 - 12 = 52 $
Simplify $ \sqrt{52} = 2\sqrt{13} $

## Step 3: Substitute into the quadratic formula
$ p = \frac{-8 \pm 2\sqrt{13}}{2(1)} = \frac{-8 \pm 2\sqrt{13}}{2} = -4 \pm \sqrt{13} $ (divided numerator and denominator by 2)

# Answer:
The solutions are $ p = -4 + \sqrt{13} $ and $ p = -4 - \sqrt{13} $

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QUESTION IMAGE

Question

Explanation:

Step 1: Rearrange the equation to standard quadratic form \( ax^2 + bx + c = 0 \)

Step 2: Use the quadratic formula \( v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step 3: Substitute into the quadratic formula

Step 1: Rearrange the equation to standard quadratic form \( ax^2 + bx + c = 0 \)

Step 2: Use the quadratic formula \( b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step 3: Substitute into the quadratic formula

Step 1: Rearrange the equation to standard quadratic form \( ax^2 + bx + c = 0 \)

Step 2: Use the quadratic formula \( b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step 3: Substitute into the quadratic formula

Answer: