Factoring Perfect Square Trinomial1. x²+6x+92. x²-22x+1213. x²-2x+14. x²+20x+1005. x²+24x+1446. 36x²-60x+257. 4x²-36x+818. x²-24x+144(Note: Please explain in a step by step formula)​

Step-by-step explanation:Factoring a perfect square trinomial involves recognizing that the expression can be written in the form \((a + b)^2\) or \((a - b)^2\). The general formula for a perfect square trinomial is:- If \((a + b)^2 = a^2 + 2ab + b^2\)- If \((a - b)^2 = a^2 - 2ab + b^2\)### Steps to Factor a Perfect Square Trinomial1. **Identify the quadratic trinomial**: It will typically be in the form \(x^2 + bx + c\).2. **Check if it can be expressed as a perfect square**: - The \(c\) term should equal the square of half the \(b\) term. - \((\frac{b}{2})^2 = c\)3. **Factor it into the appropriate square form**: - If the middle term \(b\) is positive, use \((x + \frac{b}{2})^2\). - If the middle term \(b\) is negative, use \((x - \frac{|b|}{2})^2\).### Let's factor the given expressions:1. **\(x^2 + 6x + 9\)**: - \(b = 6\), \(c = 9\) - \((\frac{6}{2})^2 = 3^2 = 9\) (True) - Factor: \((x + 3)^2\)2. **\(x^2 - 22x + 121\)**: - \(b = -22\), \(c = 121\) - \((\frac{-22}{2})^2 = (-11)^2 = 121\) (True) - Factor: \((x - 11)^2\)3. **\(x^2 - 2x + 1\)**: - \(b = -2\), \(c = 1\) - \((\frac{-2}{2})^2 = (-1)^2 = 1\) (True) - Factor: \((x - 1)^2\)4. **\(x^2 + 20x + 100\)**: - \(b = 20\), \(c = 100\) - \((\frac{20}{2})^2 = 10^2 = 100\) (True) - Factor: \((x + 10)^2\)5. **\(x^2 + 24x + 144\)**: - \(b = 24\), \(c = 144\) - \((\frac{24}{2})^2 = 12^2 = 144\) (True) - Factor: \((x + 12)^2\)6. **\(36x^2 - 60x + 25\)**: - Rewrite as \((6x)^2 - 2(6x)(\frac{5}{6}) + (\frac{5}{6})^2\) - Confirm \(b = -10\) and \(c = 25\) for \((\frac{-10}{2})^2 = 5^2 = 25\) (True) - Factor: \((6x - 5)^2\)7. **\(4x^2 - 36x + 81\)**: - Rewrite as \((2x)^2 - 2(2x)(9) + 9^2\) - Confirm \(b = -18\) and \(c = 81\) for \((\frac{-18}{2})^2 = 9^2 = 81\) (True) - Factor: \((2x - 9)^2\)8. **\(x^2 - 24x + 144\)**: - \(b = -24\), \(c = 144\) - \((\frac{-24}{2})^2 = (-12)^2 = 144\) (True) - Factor: \((x - 12)^2\)### Summary of the Factors:1. \(x^2 + 6x + 9 = (x + 3)^2\)2. \(x^2 - 22x + 121 = (x - 11)^2\)3. \(x^2 - 2x + 1 = (x - 1)^2\)4. \(x^2 + 20x + 100 = (x + 10)^2\)5. \(x^2 + 24x + 144 = (x + 12)^2\)6. \(36x^2 - 60x + 25 = (6x - 5)^2\)7. \(4x^2 - 36x + 81 = (2x - 9)^2\)8. \(x^2 - 24x + 144 = (x - 12)^2\)

Answered by romnickpallon | 2024-09-05