Factor \( 8x^2 - 18x + 9 \). First, check if it is factorable by looking for two numbers that multiply to \( 8 imes 9 = 72 \) and add to \( -18 \). These numbers are \( -12 \) and \( -6 \).

["## Factoring the Quadratic ( 8x^2 - 18x + 9 ): Is It Factoreable?", "When tackling quadratic expressions like ( 8x^2 - 18x + 9 ), one of the first questions you should ask is: Is this quadratic factorable? Understanding whether an expression can be factored is crucial for solving equations, simplifying expressions, and working with functions.", "### Can ( 8x^2 - 18x + 9 ) Be Factored?", "To determine factorability, we examine the coefficients:\n- Leading coefficient = 8\n- Constant term = 9\n- Middle coefficient = -18", "A helpful method is to check if we can find two numbers that:\n1. Multiply to ( a \cdot c = 8 \ imes 9 = 72 ), and\n2. Add to the middle coefficient ( -18 ).", "We’re looking for two numbers that multiply to 72 and add to -18.", "### Finding Suitable Numbers", "Let’s list factor pairs of 72 and check their sums:", "- ( 1 \ imes 72 ): ( 1 + 72 = 73 )\n- ( 2 \ imes 36 ): ( 2 + 36 = 38 )\n- ( 3 \ imes 24 ): ( 3 + 24 = 27 )\n- ( 4 \ imes 18 ): ( 4 + 18 = 22 )\n- ( 6 \ imes 12 ): ( 6 + 12 = 18 )", "Negative pairs:", "- ( -6 \ imes -12 ): ( (-6) + (-12) = -18 ) ✅", "Perfect! We found two numbers: -6 and -12, which multiply to 72 and add to -18.", "### Rewriting the Middle Term", "Now use these numbers to split the middle term:", "[\n8x^2 - 18x + 9 = 8x^2 - 12x - 6x + 9\n]", "Next, factor by grouping:", "[\n= (8x^2 - 12x) + (-6x + 9)\n]\n[\n= 4x(2x - 3) - 3(2x - 3)\n]", "### Completing the Factorization", "Factor out the common binomial ( (2x - 3) ):", "[\n= (2x - 3)(4x - 3)\n]", "---", "### Final Answer", "[\n8x^2 - 18x + 9 = (2x - 3)(4x - 3)\n]", "---", "### Why This Factoring Matters", "Factoring expressions like ( 8x^2 - 18x + 9 ) plays a vital role in:", "- Solving quadratic equations using the zero product property\n- Simplifying rational expressions\n- Analyzing quadratic functions (like finding roots and vertex)\n- Teaching fundamental algebra skills", "Understanding whether and how to factor quadratics equips you with a powerful tool for working with polynomials and solving real-world math problems.", "---", "### Key Takeaways", "- Factorability is confirmed when ( a \cdot c = \Delta ) (product) and two numbers sum to the middle coefficient.\n- Here, ( 72 ) splits into -6 and -12, enabling the split and grouping method.\n- The fully factored form is ( (2x - 3)(4x - 3) ).", "---", "Keywords for SEO:\nFactor ( 8x^2 - 18x + 9 ), factor quadratic expression, factor ( 8x^2 - 18x + 9 ), factor by grouping, quadratic factoring, algebra techniques, solving quadratics, polynomial factorization", "---", "### Additional Notes", "If discriminant ( b^2 - 4ac ) is negative, factoring over real numbers isn’t possible. Here:", "[\n(-18)^2 - 4(8)(9) = 324 - 288 = 36 > 0\n]", "Since discriminant is positive, there are two real roots—and our factorization confirms this.", "Understanding the conditions behind factoring helps deepen algebra fluency and prepares you for advanced math topics!"]