name:
date: per: #1,3,5,7,9,11
unit 4: solving quadratic equations
homework 8: the quadratic formula
** this is a 2 - page document! **
directions: solve each equation by the quadratic formula.
1. $x^{2}+12x - 8 = 0$
2. $-2x^{2}+7x = 3$
3. $-x^{2}+5x - 6 = 2$
4. $13x^{2}-16x = x^{2}-x$
5. $8x^{2}+15x = 7x - 4$
6. $-x^{2}+18 = 93$
© gina wilson (all things algebra®, llc), 2015 - 2022

Question

name:
date: 	per: 	#1,3,5,7,9,11
unit 4: solving quadratic equations
homework 8: the quadratic formula
** this is a 2 - page document! **
directions: solve each equation by the quadratic formula.
1. $x^{2}+12x - 8 = 0$
2. $-2x^{2}+7x = 3$
3. $-x^{2}+5x - 6 = 2$
4. $13x^{2}-16x = x^{2}-x$
5. $8x^{2}+15x = 7x - 4$
6. $-x^{2}+18 = 93$
© gina wilson (all things algebra®, llc), 2015 - 2022

Accepted Answer

### Problem 1: $ x^2 + 12x - 8 = 0 $
# Explanation:
## Step 1: Identify $ a $, $ b $, $ c $
For a quadratic equation $ ax^2 + bx + c = 0 $, here $ a = 1 $, $ b = 12 $, $ c = -8 $.
## Step 2: Apply Quadratic Formula
The quadratic formula is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. Substitute $ a = 1 $, $ b = 12 $, $ c = -8 $:
$ x = \frac{-12 \pm \sqrt{12^2 - 4(1)(-8)}}{2(1)} $
## Step 3: Simplify the discriminant
Calculate $ 12^2 - 4(1)(-8) = 144 + 32 = 176 $. So $ x = \frac{-12 \pm \sqrt{176}}{2} $.
Simplify $ \sqrt{176} = \sqrt{16 \times 11} = 4\sqrt{11} $. Then $ x = \frac{-12 \pm 4\sqrt{11}}{2} = -6 \pm 2\sqrt{11} $.
# Answer: $ x = -6 + 2\sqrt{11} $ or $ x = -6 - 2\sqrt{11} $

### Problem 2: $ -2x^2 + 7x = 3 $
# Explanation:
## Step 1: Rewrite in standard form
Subtract 3 from both sides: $ -2x^2 + 7x - 3 = 0 $. Here $ a = -2 $, $ b = 7 $, $ c = -3 $.
## Step 2: Apply Quadratic Formula
$ x = \frac{-7 \pm \sqrt{7^2 - 4(-2)(-3)}}{2(-2)} $
## Step 3: Simplify the discriminant
Calculate $ 7^2 - 4(-2)(-3) = 49 - 24 = 25 $. So $ x = \frac{-7 \pm \sqrt{25}}{-4} = \frac{-7 \pm 5}{-4} $.
Two solutions: $ x = \frac{-7 + 5}{-4} = \frac{-2}{-4} = \frac{1}{2} $ and $ x = \frac{-7 - 5}{-4} = \frac{-12}{-4} = 3 $.
# Answer: $ x = \frac{1}{2} $ or $ x = 3 $

### Problem 3: $ -x^2 + 5x - 6 = 2 $
# Explanation:
## Step 1: Rewrite in standard form
Subtract 2 from both sides: $ -x^2 + 5x - 8 = 0 $, multiply by -1: $ x^2 - 5x + 8 = 0 $. Here $ a = 1 $, $ b = -5 $, $ c = 8 $.
## Step 2: Apply Quadratic Formula
$ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(8)}}{2(1)} = \frac{5 \pm \sqrt{25 - 32}}{2} $
## Step 3: Analyze the discriminant
$ 25 - 32 = -7 $, so $ \sqrt{-7} $ is imaginary. The solutions are $ x = \frac{5 \pm i\sqrt{7}}{2} $ (complex numbers).
# Answer: $ x = \frac{5 + i\sqrt{7}}{2} $ or $ x = \frac{5 - i\sqrt{7}}{2} $

### Problem 4: $ 13x^2 - 16x = x^2 - x $
# Explanation:
## Step 1: Rewrite in standard form
Subtract $ x^2 - x $ from both sides: $ 12x^2 - 15x = 0 $. Here $ a = 12 $, $ b = -15 $, $ c = 0 $.
## Step 2: Apply Quadratic Formula
$ x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(12)(0)}}{2(12)} = \frac{15 \pm \sqrt{225}}{24} $
## Step 3: Simplify
$ \sqrt{225} = 15 $, so $ x = \frac{15 \pm 15}{24} $. Two solutions: $ x = \frac{15 + 15}{24} = \frac{30}{24} = \frac{5}{4} $ and $ x = \frac{15 - 15}{24} = 0 $.
# Answer: $ x = 0 $ or $ x = \frac{5}{4} $

### Problem 5: $ 8x^2 + 15x = 7x - 4 $
# Explanation:
## Step 1: Rewrite in standard form
Subtract $ 7x - 4 $ from both sides: $ 8x^2 + 8x + 4 = 0 $, divide by 4: $ 2x^2 + 2x + 1 = 0 $. Here $ a = 2 $, $ b = 2 $, $ c = 1 $.
## Step 2: Apply Quadratic Formula
$ x = \frac{-2 \pm \sqrt{2^2 - 4(2)(1)}}{2(2)} = \frac{-2 \pm \sqrt{4 - 8}}{4} $
## Step 3: Analyze the discriminant
$ 4 - 8 = -4 $, so $ \sqrt{-4} = 2i $. Solutions: $ x = \frac{-2 \pm 2i}{4} = \frac{-1 \pm i}{2} $ (complex numbers).
# Answer: $ x = \frac{-1 + i}{2} $ or $ x = \frac{-1 - i}{2} $

### Problem 6: $ -x^2 + 18 = 93 $
# Explanation:
## Step 1: Rewrite in standard form
Subtract 93 from both sides: $ -x^2 - 75 = 0 $, multiply by -1: $ x^2 + 75 = 0 $. Here $ a = 1 $, $ b = 0 $, $ c = 75 $.
## Step 2: Apply Quadratic Formula
$ x = \frac{-0 \pm \sqrt{0^2 - 4(1)(75)}}{2(1)} = \frac{\pm \sqrt{-300}}{2} $
## Step 3: Simplify
$ \sqrt{-300} = 10i\sqrt{3} $, so $ x = \frac{\pm 10i\sqrt{3}}{2} = \pm 5i\sqrt{3} $ (complex numbers).
# Answer: $ x = 5i\sqrt{3} $ or $ x = -5i\sqrt{3} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: \( x^2 + 12x - 8 = 0 \)

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

Step 2: Apply Quadratic Formula

Step 3: Simplify the discriminant

Step 1: Rewrite in standard form

Step 2: Apply Quadratic Formula

Step 3: Simplify the discriminant

Step 1: Rewrite in standard form

Step 2: Apply Quadratic Formula

Step 3: Analyze the discriminant

Answer:

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