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. \( (14 x-21)(-8 x+21) \)

Solve the equation by following steps: - step0: Solve for \(x\): \(x^{2}-225=\left(x-15\right)\left(x+15\right)\) - step1: Move the expression to the left side: \(x^{2}-225-\left(x-15\right)\left(x+15\right)=0\) - 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 \(x\): \(x^{2}-36x+324=\left(x-18\right)\left(x+18\right)\) - step1: Move the expression to the left side: \(x^{2}-36x+324-\left(x-18\right)\left(x+18\right)=0\) - 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 \(x\): \(9\left(x+7\right)^{2}-121x^{2}=\left(14x-21\right)\left(-8x+21\right)\) - step1: Calculate: \(-112x^{2}+126x+441=\left(14x-21\right)\left(-8x+21\right)\) - step2: Calculate: \(-112x^{2}+126x+441=-112x^{2}+462x-441\) - step3: Move the expression to the left side: \(-112x^{2}+126x+441-\left(-112x^{2}+462x-441\right)=0\) - 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 \(x\): \(x^{2}-81=\left(x-9\right)^{2}\) - 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: \(144x^{2}-\left(5x-3\right)^{2}=\left(12x-3\right)\left(17x+3\right)\) - 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: \(-5x\left(17x-9\right)=0\) - 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 \(x\): \(x^{2}-81=\left(x+9\right)^{2}\) - 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 \(x\): \(x^{2}-81=\left(x+9\right)\left(x-9\right)\) - step1: Move the expression to the left side: \(x^{2}-81-\left(x+9\right)\left(x-9\right)=0\) - 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 \(x\): \(x^{2}-81=\left(x+9\right)\left(x+9\right)\) - step1: Multiply the terms: \(x^{2}-81=\left(x+9\right)^{2}\) - 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: \(144x^{2}-\left(5x-3\right)^{2}=\left(12x-3\right)\left(12x+3\right)\) - 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: \(-5x\left(5x-6\right)=0\) - 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 \(x\): \(9\left(x+7\right)^{2}-121x^{2}=\left(14x+7\right)\left(-8x+7\right)\) - step1: Calculate: \(-112x^{2}+126x+441=\left(14x+7\right)\left(-8x+7\right)\) - step2: Calculate: \(-112x^{2}+126x+441=-112x^{2}+42x+49\) - step3: Move the expression to the left side: \(-112x^{2}+126x+441-\left(-112x^{2}+42x+49\right)=0\) - 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 \(x\): \(9\left(x+7\right)^{2}-121x^{2}=\left(14x+21\right)\left(-8x+21\right)\) - step1: Calculate: \(-112x^{2}+126x+441=\left(14x+21\right)\left(-8x+21\right)\) - step2: Calculate: \(-112x^{2}+126x+441=-112x^{2}+126x+441\) - step3: Move the expression to the left side: \(-112x^{2}+126x+441-\left(-112x^{2}+126x+441\right)=0\) - 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 \(x\): \(144x^{2}-\left(5x-3\right)^{2}=\left(17x-3\right)\left(7x-3\right)\) - step1: Calculate: \(119x^{2}+30x-9=\left(17x-3\right)\left(7x-3\right)\) - step2: Calculate: \(119x^{2}+30x-9=119x^{2}-72x+9\) - step3: Move the expression to the left side: \(119x^{2}+30x-9-\left(119x^{2}-72x+9\right)=0\) - 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 \(x\): \(x^{2}-169=\left(x-13\right)^{2}\) - 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 \(x\): \(x^{2}-36=\left(x+36\right)\left(x-36\right)\) - step1: Move the expression to the left side: \(x^{2}-36-\left(x+36\right)\left(x-36\right)=0\) - 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 \(x\): \(144x^{2}-\left(5x-3\right)^{2}=\left(17x-3\right)\left(7x+3\right)\) - step1: Calculate: \(119x^{2}+30x-9=\left(17x-3\right)\left(7x+3\right)\) - step2: Calculate: \(119x^{2}+30x-9=119x^{2}+30x-9\) - step3: Move the expression to the left side: \(119x^{2}+30x-9-\left(119x^{2}+30x-9\right)=0\) - 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: \(9\left(x+7\right)^{2}-121x^{2}=\left(14x+7\right)\left(8x+7\right)\) - step1: Expand the expression: \(-112x^{2}+126x+441=\left(14x+7\right)\left(8x+7\right)\) - 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: \(x=\frac{-28\pm \sqrt{28^{2}-4\times 224\left(-392\right)}}{2\times 224}\) - step6: Simplify the expression: \(x=\frac{-28\pm \sqrt{28^{2}-4\times 224\left(-392\right)}}{448}\) - 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