20. ข้อใดคือการแยกตัวประกอบของพหุนามที่ถูกต้อง ก. $$ x^{2}-36=(x+36)(x-36) $$ ข. $$ x^{2}-225=(x-15)(x+15) $$ x $$ x^{2}-169=(x-13)^{2} $$ ง. $$ x^{2}-36 x+324=(x-18)(x+18) $$ 21. ข้อใดคือการแยกตัวประกอบของ $$ x^{2}-81 $$ ก. $$ (x-9)^{2} $$ 8. $$ (x+9)^{2} $$ ค. $$ (x+9)(x-9) $$ ง. $$ (x+9)(x+9) $$ 22. ข้อใดคือการแยกตัวประกอบของ $$ 144 x^{2}-(5 x-3)^{2} $$ $$ (12 x-3)(12 x+3) $$ ข. $$ (12 x-3)(17 x+3) $$ ค. $$ (17 x-3)(7 x-3) $$ X $$ (17 x-3)(7 x+3) $$ 23. ข้อใดคือการแยกตัวประกอบของ $$ 9(x+7)^{2}-121 x^{2} $$ $$ x(14 x+7)(-8 x+7) $$ ข. $$ (14 x+21)(-8 x+21) $$ ค. $$ (14 x+7)(8 x+7) $$ ง. $$ (14 x-21)(-8 x+21) $$

Question

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-225=\left(x-15ight)\left(x+15ight)$$
- step1: Move the expression to the left side:
  $$x^{2}-225-\left(x-15ight)\left(x+15ight)=0$$
- step2: Subtract the terms:
  $$0=0$$
- step3: The statement is true:
  $$x \in \mathbb{R}$$
Solve the equation $$ x^{2}-36 x+324=(x-18)(x+18) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-36x+324=\left(x-18ight)\left(x+18ight)$$
- step1: Move the expression to the left side:
  $$x^{2}-36x+324-\left(x-18ight)\left(x+18ight)=0$$
- step2: Calculate:
  $$648-36x=0$$
- step3: Move the constant to the right side:
  $$-36x=0-648$$
- step4: Remove 0:
  $$-36x=-648$$
- step5: Change the signs:
  $$36x=648$$
- step6: Divide both sides:
  $$\frac{36x}{36}=\frac{648}{36}$$
- step7: Divide the numbers:
  $$x=18$$
Solve the equation $$ 9(x+7)^{2}-121 x^{2}=(14 x-21)(-8 x+21) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$9\left(x+7ight)^{2}-121x^{2}=\left(14x-21ight)\left(-8x+21ight)$$
- step1: Calculate:
  $$-112x^{2}+126x+441=\left(14x-21ight)\left(-8x+21ight)$$
- step2: Calculate:
  $$-112x^{2}+126x+441=-112x^{2}+462x-441$$
- step3: Move the expression to the left side:
  $$-112x^{2}+126x+441-\left(-112x^{2}+462x-441ight)=0$$
- step4: Calculate:
  $$-336x+882=0$$
- step5: Move the constant to the right side:
  $$-336x=0-882$$
- step6: Remove 0:
  $$-336x=-882$$
- step7: Change the signs:
  $$336x=882$$
- step8: Divide both sides:
  $$\frac{336x}{336}=\frac{882}{336}$$
- step9: Divide the numbers:
  $$x=\frac{21}{8}$$
Solve the equation $$ x^{2}-81=(x-9)^{2} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-81=\left(x-9ight)^{2}$$
- step1: Expand the expression:
  $$x^{2}-81=x^{2}-18x+81$$
- step2: Cancel equal terms:
  $$-81=-18x+81$$
- step3: Swap the sides:
  $$-18x+81=-81$$
- step4: Move the constant to the right side:
  $$-18x=-81-81$$
- step5: Subtract the numbers:
  $$-18x=-162$$
- step6: Change the signs:
  $$18x=162$$
- step7: Divide both sides:
  $$\frac{18x}{18}=\frac{162}{18}$$
- step8: Divide the numbers:
  $$x=9$$
Solve the equation $$ 144 x^{2}-(5 x-3)^{2}=(12 x-3)(17 x+3) $$.
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$144x^{2}-\left(5x-3ight)^{2}=\left(12x-3ight)\left(17x+3ight)$$
- step1: Expand the expression:
  $$119x^{2}+30x-9=204x^{2}-15x-9$$
- step2: Move the expression to the left side:
  $$-85x^{2}+45x=0$$
- step3: Factor the expression:
  $$-5x\left(17x-9ight)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&17x-9=0\&-5x=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&x=\frac{9}{17}\&x=0\end{align}$$
- step6: Rewrite:
  $$x_{1}=0,x_{2}=\frac{9}{17}$$
Solve the equation $$ x^{2}-81=(x+9)^{2} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-81=\left(x+9ight)^{2}$$
- step1: Expand the expression:
  $$x^{2}-81=x^{2}+18x+81$$
- step2: Cancel equal terms:
  $$-81=18x+81$$
- step3: Swap the sides:
  $$18x+81=-81$$
- step4: Move the constant to the right side:
  $$18x=-81-81$$
- step5: Subtract the numbers:
  $$18x=-162$$
- step6: Divide both sides:
  $$\frac{18x}{18}=\frac{-162}{18}$$
- step7: Divide the numbers:
  $$x=-9$$
Solve the equation $$ x^{2}-81=(x+9)(x-9) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-81=\left(x+9ight)\left(x-9ight)$$
- step1: Move the expression to the left side:
  $$x^{2}-81-\left(x+9ight)\left(x-9ight)=0$$
- step2: Subtract the terms:
  $$0=0$$
- step3: The statement is true:
  $$x \in \mathbb{R}$$
Solve the equation $$ x^{2}-81=(x+9)(x+9) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-81=\left(x+9ight)\left(x+9ight)$$
- step1: Multiply the terms:
  $$x^{2}-81=\left(x+9ight)^{2}$$
- step2: Expand the expression:
  $$x^{2}-81=x^{2}+18x+81$$
- step3: Cancel equal terms:
  $$-81=18x+81$$
- step4: Swap the sides:
  $$18x+81=-81$$
- step5: Move the constant to the right side:
  $$18x=-81-81$$
- step6: Subtract the numbers:
  $$18x=-162$$
- step7: Divide both sides:
  $$\frac{18x}{18}=\frac{-162}{18}$$
- step8: Divide the numbers:
  $$x=-9$$
Solve the equation $$ 144 x^{2}-(5 x-3)^{2}=(12 x-3)(12 x+3) $$.
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$144x^{2}-\left(5x-3ight)^{2}=\left(12x-3ight)\left(12x+3ight)$$
- step1: Expand the expression:
  $$119x^{2}+30x-9=144x^{2}-9$$
- step2: Move the expression to the left side:
  $$-25x^{2}+30x=0$$
- step3: Factor the expression:
  $$-5x\left(5x-6ight)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&5x-6=0\&-5x=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&x=\frac{6}{5}\&x=0\end{align}$$
- step6: Rewrite:
  $$x_{1}=0,x_{2}=\frac{6}{5}$$
Solve the equation $$ 9(x+7)^{2}-121 x^{2}=(14 x+7)(-8 x+7) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$9\left(x+7ight)^{2}-121x^{2}=\left(14x+7ight)\left(-8x+7ight)$$
- step1: Calculate:
  $$-112x^{2}+126x+441=\left(14x+7ight)\left(-8x+7ight)$$
- step2: Calculate:
  $$-112x^{2}+126x+441=-112x^{2}+42x+49$$
- step3: Move the expression to the left side:
  $$-112x^{2}+126x+441-\left(-112x^{2}+42x+49ight)=0$$
- step4: Calculate:
  $$84x+392=0$$
- step5: Move the constant to the right side:
  $$84x=0-392$$
- step6: Remove 0:
  $$84x=-392$$
- step7: Divide both sides:
  $$\frac{84x}{84}=\frac{-392}{84}$$
- step8: Divide the numbers:
  $$x=-\frac{14}{3}$$
Solve the equation $$ 9(x+7)^{2}-121 x^{2}=(14 x+21)(-8 x+21) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$9\left(x+7ight)^{2}-121x^{2}=\left(14x+21ight)\left(-8x+21ight)$$
- step1: Calculate:
  $$-112x^{2}+126x+441=\left(14x+21ight)\left(-8x+21ight)$$
- step2: Calculate:
  $$-112x^{2}+126x+441=-112x^{2}+126x+441$$
- step3: Move the expression to the left side:
  $$-112x^{2}+126x+441-\left(-112x^{2}+126x+441ight)=0$$
- step4: Calculate:
  $$0=0$$
- step5: The statement is true:
  $$x \in \mathbb{R}$$
Solve the equation $$ 144 x^{2}-(5 x-3)^{2}=(17 x-3)(7 x-3) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$144x^{2}-\left(5x-3ight)^{2}=\left(17x-3ight)\left(7x-3ight)$$
- step1: Calculate:
  $$119x^{2}+30x-9=\left(17x-3ight)\left(7x-3ight)$$
- step2: Calculate:
  $$119x^{2}+30x-9=119x^{2}-72x+9$$
- step3: Move the expression to the left side:
  $$119x^{2}+30x-9-\left(119x^{2}-72x+9ight)=0$$
- step4: Calculate:
  $$102x-18=0$$
- step5: Move the constant to the right side:
  $$102x=0+18$$
- step6: Remove 0:
  $$102x=18$$
- step7: Divide both sides:
  $$\frac{102x}{102}=\frac{18}{102}$$
- step8: Divide the numbers:
  $$x=\frac{3}{17}$$
Solve the equation $$ x^{2}-169=(x-13)^{2} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-169=\left(x-13ight)^{2}$$
- step1: Expand the expression:
  $$x^{2}-169=x^{2}-26x+169$$
- step2: Cancel equal terms:
  $$-169=-26x+169$$
- step3: Swap the sides:
  $$-26x+169=-169$$
- step4: Move the constant to the right side:
  $$-26x=-169-169$$
- step5: Subtract the numbers:
  $$-26x=-338$$
- step6: Change the signs:
  $$26x=338$$
- step7: Divide both sides:
  $$\frac{26x}{26}=\frac{338}{26}$$
- step8: Divide the numbers:
  $$x=13$$
Solve the equation $$ x^{2}-36=(x+36)(x-36) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-36=\left(x+36ight)\left(x-36ight)$$
- step1: Move the expression to the left side:
  $$x^{2}-36-\left(x+36ight)\left(x-36ight)=0$$
- step2: Subtract the terms:
  $$1260=0$$
- step3: The statement is false:
  $$x \in \varnothing $$
Solve the equation $$ 144 x^{2}-(5 x-3)^{2}=(17 x-3)(7 x+3) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$144x^{2}-\left(5x-3ight)^{2}=\left(17x-3ight)\left(7x+3ight)$$
- step1: Calculate:
  $$119x^{2}+30x-9=\left(17x-3ight)\left(7x+3ight)$$
- step2: Calculate:
  $$119x^{2}+30x-9=119x^{2}+30x-9$$
- step3: Move the expression to the left side:
  $$119x^{2}+30x-9-\left(119x^{2}+30x-9ight)=0$$
- step4: Calculate:
  $$0=0$$
- step5: The statement is true:
  $$x \in \mathbb{R}$$
Solve the equation $$ 9(x+7)^{2}-121 x^{2}=(14 x+7)(8 x+7) $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$9\left(x+7ight)^{2}-121x^{2}=\left(14x+7ight)\left(8x+7ight)$$
- step1: Expand the expression:
  $$-112x^{2}+126x+441=\left(14x+7ight)\left(8x+7ight)$$
- step2: Expand the expression:
  $$-112x^{2}+126x+441=112x^{2}+154x+49$$
- step3: Move the expression to the left side:
  $$-224x^{2}-28x+392=0$$
- step4: Multiply both sides:
  $$224x^{2}+28x-392=0$$
- step5: Solve using the quadratic formula:
  $$x=\frac{-28\pm \sqrt{28^{2}-4	imes 224\left(-392ight)}}{2	imes 224}$$
- step6: Simplify the expression:
  $$x=\frac{-28\pm \sqrt{28^{2}-4	imes 224\left(-392ight)}}{448}$$
- step7: Simplify the expression:
  $$x=\frac{-28\pm \sqrt{352016}}{448}$$
- step8: Simplify the expression:
  $$x=\frac{-28\pm 28\sqrt{449}}{448}$$
- step9: Separate into possible cases:
  $$\begin{align}&x=\frac{-28+28\sqrt{449}}{448}\&x=\frac{-28-28\sqrt{449}}{448}\end{align}$$
- step10: Simplify the expression:
  $$\begin{align}&x=\frac{-1+\sqrt{449}}{16}\&x=\frac{-28-28\sqrt{449}}{448}\end{align}$$
- step11: Simplify the expression:
  $$\begin{align}&x=\frac{-1+\sqrt{449}}{16}\&x=-\frac{1+\sqrt{449}}{16}\end{align}$$
- step12: Rewrite:
  $$x_{1}=-\frac{1+\sqrt{449}}{16},x_{2}=\frac{-1+\sqrt{449}}{16}$$
เรามาแยกตัวประกอบของพหุนามแต่ละข้อกันทีละข้อ:

### ข้อ 20
1. **ก.** $$ x^{2}-36=(x+36)(x-36) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ x^{2}-36 = (x-6)(x+6) $$

2. **ข.** $$ x^{2}-225=(x-15)(x+15) $$ 
   - **ถูกต้อง** เพราะ $$ x^{2}-225 = (x-15)(x+15) $$

3. **ค.** $$ x^{2}-169=(x-13)^{2} $$ 
   - **ไม่ถูกต้อง** เพราะ $$ x^{2}-169 = (x-13)(x+13) $$

4. **ง.** $$ x^{2}-36 x+324=(x-18)(x+18) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ x^{2}-36x+324 = (x-18)^{2} $$

### ข้อ 21
1. **ก.** $$ (x-9)^{2} $$ 
   - **ไม่ถูกต้อง** เพราะ $$ x^{2}-81 = (x-9)(x+9) $$

2. **ข.** $$ (x+9)^{2} $$ 
   - **ไม่ถูกต้อง** เพราะ $$ x^{2}-81 = (x-9)(x+9) $$

3. **ค.** $$ (x+9)(x-9) $$ 
   - **ถูกต้อง** เพราะ $$ x^{2}-81 = (x-9)(x+9) $$

4. **ง.** $$ (x+9)(x+9) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ x^{2}-81 = (x-9)(x+9) $$

### ข้อ 22
1. **ก.** $$ (12 x-3)(12 x+3) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 144 x^{2}-(5 x-3)^{2} = (12x-5)(12x+5) $$

2. **ข.** $$ (12 x-3)(17 x+3) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 144 x^{2}-(5 x-3)^{2} = (12x-5)(12x+5) $$

3. **ค.** $$ (17 x-3)(7 x-3) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 144 x^{2}-(5 x-3)^{2} = (12x-5)(12x+5) $$

4. **ง.** $$ (17 x-3)(7 x+3) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 144 x^{2}-(5 x-3)^{2} = (12x-5)(12x+5) $$

### ข้อ 23
1. **ก.** $$ x(14 x+7)(-8 x+7) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 9(x+7)^{2}-121 x^{2} = (14x-21)(-8x+21) $$

2. **ข.** $$ (14 x+21)(-8 x+21) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 9(x+7)^{2}-121 x^{2} = (14x-21)(-8x+21) $$

3. **ค.** $$ (14 x+7)(8 x+7) $$ 
   - **ไม่ถูกต้อง** เพราะ $$ 9(x+7)^{2}-121 x^{2} = (14x-21)(-8x+21) $$

4. **ง.** $$ (14 x-21)(-8 x+21) $$ 
   - **ถูกต้อง** เพราะ $$ 9(x+7)^{2}-121 x^{2} = (14x-21)(-8x+21) $$

### สรุป
- ข้อ 20: ข. ถูกต้อง
- ข้อ 21: ค. ถูกต้อง
- ข้อ 22: ไม่มีข้อถูกต้อง
- ข้อ 23: ง. ถูกต้อง
- ข้อ 20: ข. $$ x^{2}-225=(x-15)(x+15) $$ ถูกต้อง - ข้อ 21: ค. $$ (x+9)(x-9) $$ ถูกต้อง - ข้อ 22: ไม่มีข้อถูกต้อง - ข้อ 23: ง. $$ (14x-21)(-8x+21) $$ ถูกต้อง

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