\boxed{10a^2 + 29ab - 21b^2}

["Understanding the Quadratic Expression: (10a^2 + 29ab - 21b^2)", "The expression (10a^2 + 29ab - 21b^2) is a quadratic form in two variables, often studied in algebra and applied mathematics. Whether you’re solving equations, optimizing functions, or analyzing relationships between variables, understanding this expression unlocks powerful tools in mathematical modeling and algebra.", "---", "### What Is (10a^2 + 29ab - 21b^2)?", "This expression is a second-degree polynomial in variables (a) and (b). It combines squared terms (a^2) and (b^2), along with a cross term (ab) with a positive and negative coefficient, forming a generalized quadratic. Designed in the standard form:", "[\nQ(a, b) = 10a^2 + 29ab - 21b^2\n]", "This structure makes it particularly useful in contexts like conic sections, systems of equations, and multivariate calculus.", "---", "### Why Study This Quadratic?", "Understanding quadratic expressions like (10a^2 + 29ab - 21b^2) is essential for:", "- Factorization and Root Analysis: Determining when (Q(a,b) = 0) helps solve linear or simultaneous equations, especially in geometric and optimization problems.\n- Graphing Behavior: The expression defines a quadratic curve, which, depending on the discriminant, may be hyperbolic or parabolic in shape—key in optimization and contour plotting.\n- Applications in Real-World Modeling: In economics, physics, and engineering, such forms model interactions between two variables (like cost vs. volume, force components, or growth rates).", "---", "### Factoring (10a^2 + 29ab - 21b^2)", "Step 1: Treat it as a quadratic in (a), with (b) as a constant:", "[\nQ(a,b) = 10a^2 + (29b)a - 21b^2\n]", "Use the quadratic formula to factor:", "[\na = \frac{-(29b) \pm \sqrt{(29b)^2 - 4(10)(-21b^2)}}{2(10)}\n]", "Calculate the discriminant:", "[\n(29b)^2 + 840b^2 = 841b^2 + 840b^2 = 1681b^2\n]", "[\n\sqrt{1681b^2} = 41b \quad (\ ext{since } \sqrt{1681} = 41 \ ext{ and } b^2 \ ext{ gives } |b|, \ ext{ but assume } b > 0 \ ext{ for simplicity})\n]", "Thus,", "[\na = \frac{-29b \pm 41b}{20}\n]", "So the two solutions are:", "[\na = \frac{12b}{20} = \frac{3b}{5}, \quad a = \frac{-70b}{20} = -\frac{7b}{2}\n]", "Step 2: Write the factored form:", "[\n10a^2 + 29ab - 21b^2 = 10\left(a - \frac{3}{5}b\right)\left(a + \frac{7}{2}b\right)\n]", "> 🔍 Note: To avoid fractions, expand back or use integer coefficients with multiplication by 10 (common in factoring):", "[\nQ(a,b) = 10\left(a - \frac{3}{5}b\right)(a + \frac{7}{2}b) \quad \ ext{or} \quad Q(a,b) = (10a - 6b)(2a + 7b)\n]", "Check:", "[\n(10a - 6b)(2a + 7b) = 20a^2 + 70ab - 12ab - 42b^2 = 20a^2 + 58ab - 42b^2\n]", "Oops! Sign mismatch — the original discriminant gave (+41b) and (-7b/2), but the sign flipped. Correcting:", "Actually:", "[\na = \frac{-29b \pm 41b}{20} \Rightarrow a = \frac{12b}{20} = \frac{3b}{5},\quad a = \frac{-70b}{20} = -\frac{7b}{2}\n]", "Thus,", "[\nQ(a,b) = 10\left(a - \frac{3}{5}b\right)\left(a + \frac{7}{2}b\right)\n]", "Multiply through to eliminate denominators:", "[\nQ(a,b) = 10 \cdot \left(a - \frac{3}{5}b\right)\left(a + \frac{7}{2}b\right) = (10a - 6b)(2a + 7b)\n]", "Verify:", "[\n(10a - 6b)(2a + 7b) = 20a^2 + 70ab - 12ab - 42b^2 = 20a^2 + 58ab - 42b^2 \quad \ ext{– still off.}\n]", "Let’s scale properly: general form of factored quadratic:", "If a quadratic (Ax^2 + Bxy + Cy^2 = m(x - \alpha y)(x - \beta y)), then coefficients must match.", "Instead, use known factorization:", "From earlier:", "[\na = \frac{-29b \pm 41b}{20} \Rightarrow \Delta = 1681b^2 = (41b)^2\n]", "So:", "[\nQ(a,b) = 10(a - \ frac{3}{5}b)(a + \ frac{7}{2}b)\n]", "To write with integer coefficients, multiply and divide:", "[\nQ(a,b) = (10a - 6b)(2a + 7b)\n]", "Wait — verify:", "[\n(10a - 6b)(2a + 7b) = 20a^2 + 70ab - 12ab - 42b^2 = 20a^2 + 58ab - 42b^2\n]", "But original: (10a^2 + 29ab - 21b^2)", "Ratio: 20 vs 10 — half. So scale down:", "Try:\n[\nQ(a,b) = 5(2a - \ frac{3}{1}b) \cdot (2a + \ frac{7}{2}b) \quad \ ext{complicated}\n]", "Better: Factor directly using leading coefficient:", "Use AC method:", "Multiply (10 \ imes (-21) = -210), find two numbers multiplying to (-210) and adding to (29):", "(35) and (-6): (35 \ imes (-6) = -210), (35 - 6 = 29)", "Break middle term:", "[\n10a^2 + 35ab - 6ab - 21b^2\n]", "Group:", "[\n(10a^2 + 35ab) + (-6ab - 21b^2) = 5a(2a + 7b) - 3b(2a + 7b) = (5a - 3b)(2a + 7b)\n]", "✅ Correct!", "---", "### Factored Form:", "[\n\boxed{10a^2 + 29ab - 21b^2 = (5a - 3b)(2a + 7b)}\n]", "---", "### Graphing the Quadratic", "The expression factors into two distinct linear terms, indicating the curve it represents is a hyperbola oriented along the lines (5a - 3b = 0) and (2a + 7b = 0). Since one factor is positive and the other negative depending on region, the graph consists of two intersecting branches.", "---", "### Application: Solving Linear Systems", "Suppose you have a system:", "[\n\begin{cases}\n(5a - 3b)(2a + 7b) = 0 \\n\ ext{(some other condition)}\n\end{cases}\n]", "This equation defines two lines; intersections with other constraints yield solution sets — useful in optimization and geometry.", "---", "### Quadratic Forms and Positivity Definiteness", "The expression (10a^2 + 29ab - 21b^2) defines a quadratic form. Its matrix representation is:", "[\nQ(\mathbf{x}) = \mathbf{x}^T A \mathbf{x},\quad A = \begin{bmatrix} 10 & 29/2 \ 29/2 & -21 \end{bmatrix}\n]", "Eigenvalues determine definiteness. Since one positive and one negative eigenvalue (from different factor signs), the form is indefinite, confirming a hyperbolic nature.", "---", "### Final Thoughts", "The quadratic expression (10a^2 + 29ab - 21b^2 = (5a - 3b)(2a + 7b)) is more than an algebraic form — it’s a gateway to analyzing relationships in two-dimensional space, solving systems, and modeling effects where variables interact multiplicatively. Mastering such factorization and interpretation strengthens skills in algebra, calculus, and applied mathematics.", "---", "### Further Reading", "- Factoring Quadratics with Two Variables\n- Conic Sections and Conic Equations\n- Quadratic Forms in Linear Algebra\n- Applications of Bivariate Polynomials in Real-World Modeling", "---", "Keywords: (10a^2 + 29ab - 21b^2), factoring quadratic expressions, binomial factoring, hyperbolic curves, quadratic forms, algebraic factoring, AP militant math mobile", "Meta Description:\nExplore the factored form ((5a - 3b)(2a + 7b)) of (10a^2 + 29ab - 21b^2), learn its significance in algebra, hyperbolic geometry, and applications in mathematics and science."]