Question

solving absolute value equations
date:
solve the following. check for extraneous solutions.

1. |6 - 2w| = 20
2. \\(\frac{|r - 5|}{3}\\) = 3
3. 4|2 - b| + 1 = -3
4. |8x - 9| - 3 = 7x

solving quadratic equations
date:
solve the following. write all irrational and complex answers in simplest form.

1. 9d² + 8 = -41
2. 2y² + 12 = -14y
3. v² - 14v + 97 = 0
4. 5r² + 2r + 1 = 0

Explanation:

Solving Absolute Value Equations

Step1: Split absolute value case 1

$6-2w=20$

Step2: Isolate $w$ (case1)

$-2w=20-6=14 \implies w=\frac{14}{-2}=-7$

Step3: Split absolute value case 2

$6-2w=-20$

Step4: Isolate $w$ (case2)

$-2w=-20-6=-26 \implies w=\frac{-26}{-2}=13$

Step5: Verify solutions

For $w=-7$: $|6-2(-7)|=|6+14|=20$, valid.
For $w=13$: $|6-2(13)|=|6-26|=20$, valid.

---

Step1: Eliminate denominator

$|r-5|=3\times3=9$

Step2: Split absolute value case1

$r-5=9$

Step3: Isolate $r$ (case1)

$r=9+5=14$

Step4: Split absolute value case2

$r-5=-9$

Step5: Isolate $r$ (case2)

$r=-9+5=-4$

Step6: Verify solutions

For $r=14$: $\frac{|14-5|}{3}=\frac{9}{3}=3$, valid.
For $r=-4$: $\frac{|-4-5|}{3}=\frac{9}{3}=3$, valid.

---

Step1: Isolate absolute value term

$4|2-b|=-3-1=-4$

Step2: Simplify absolute value

$|2-b|=\frac{-4}{4}=-1$

Step3: Check validity

Absolute value cannot be negative, no solution.

---

Step1: Isolate absolute value term

$|8x-9|=7x+3$

Step2: Split absolute value case1

$8x-9=7x+3$

Step3: Isolate $x$ (case1)

$8x-7x=3+9 \implies x=12$

Step4: Split absolute value case2

$8x-9=-(7x+3)$

Step5: Isolate $x$ (case2)

$8x-9=-7x-3 \implies 15x=6 \implies x=\frac{6}{15}=\frac{2}{5}$

Step6: Verify solutions

For $x=12$: $|8(12)-9|-3=|96-9|-3=87-3=84$, $7(12)=84$, valid.
For $x=\frac{2}{5}$: $|8(\frac{2}{5})-9|-3=|\frac{16}{5}-\frac{45}{5}|-3=|-\frac{29}{5}|-3=\frac{29}{5}-3=\frac{14}{5}$, $7(\frac{2}{5})=\frac{14}{5}$, valid.

Step1: Isolate squared term

$9d^2=-41-8=-49$

Step2: Solve for $d^2$

$d^2=\frac{-49}{9}$

Step3: Find complex roots

$d=\pm\sqrt{\frac{-49}{9}}=\pm\frac{7}{3}i$

---

Step1: Rearrange to standard form

$2y^2+14y+12=0$

Step2: Simplify equation

Divide by 2: $y^2+7y+6=0$

Step3: Factor quadratic

$(y+1)(y+6)=0$

Step4: Solve for $y$

$y+1=0 \implies y=-1$; $y+6=0 \implies y=-6$

---

Step1: Identify quadratic coefficients

$a=1, b=-14, c=97$

Step2: Apply quadratic formula

$v=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step3: Calculate discriminant

$b^2-4ac=(-14)^2-4(1)(97)=196-388=-192$

Step4: Simplify roots

$v=\frac{14\pm\sqrt{-192}}{2}=\frac{14\pm8\sqrt{3}i}{2}=7\pm4\sqrt{3}i$

---

Step1: Identify quadratic coefficients

$a=5, b=2, c=1$

Step2: Apply quadratic formula

$r=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step3: Calculate discriminant

$b^2-4ac=(2)^2-4(5)(1)=4-20=-16$

Step4: Simplify roots

$r=\frac{-2\pm\sqrt{-16}}{2(5)}=\frac{-2\pm4i}{10}=\frac{-1\pm2i}{5}$

Answer:

1. $w=-7$ and $w=13$
2. $r=14$ and $r=-4$
3. No solution
4. $x=12$ and $x=\frac{2}{5}$

---

Solving Quadratic Equations