Expand: \( x^2 + 2x - 3 = 8 \Rightarrow x^2 + 2x - 11 = 0 \)
["# How to Solve ( x^2 + 2x - 3 = 8 ): Step-by-Step Expansion and Solving", "Understanding how to solve quadratic equations is a fundamental skill in algebra. One common step students encounter is expanding or transforming equations like ( x^2 + 2x - 3 = 8 ) into standard form ( x^2 + 2x - 11 = 0 ). This process clarifies the equation, making it easier to apply methods such as factoring, completing the square, or using the quadratic formula.", "## Why Expand: From Word Problem to Standard Form", "In algebra, equations often start in word form or presented in non-standard arrangements, especially in applied problems. Here, ( x^2 + 2x - 3 = 8 ) is a real-world condition stating that the expression ( x^2 + 2x - 3 ) equals 8. To solve this, we move all terms to a single side to form a standard quadratic equation equal to zero.", "Step 1: Begin with the original equation:\n[\nx^2 + 2x - 3 = 8\n]", "Step 2: Subtract 8 from both sides to move the constant to the left:\n[\nx^2 + 2x - 3 - 8 = 0\n]", "Step 3: Simplify by combining constant terms:\n[\nx^2 + 2x - 11 = 0\n]", "Now the equation is transformed into standard quadratic form:\n[\nx^2 + 2x - 11 = 0\n]", "## Solving the Expanded Equation", "With the equation in standard form, ( x^2 + 2x - 11 = 0 ), several solution methods are available.", "### Factoring\nFactoring attempts to express the quadratic as the product of two binomials:\n[\nx^2 + 2x - 11 = (x + a)(x + b) = 0\n]\nWe need two numbers that multiply to (-11) and add to (2). Since 11 is prime, the pairs ( (11, -1) ) and ( (-11, 1) ) do not sum to 2. Thus, this expression does not factor nicely — meaning factoring is not straightforward here.", "### Quadratic Formula\nWhen factoring is impractical, the quadratic formula becomes the most reliable tool:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nFor ( x^2 + 2x - 11 = 0 ), the coefficients are:\n( a = 1 ), ( b = 2 ), ( c = -11 ).", "Plugging in:\n[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2}\n]", "Simplify ( \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} ):\n[\nx = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}\n]", "Solutions:\n[\nx = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}\n]", "### Graphical Interpretation\nThe original equation ( x^2 + 2x - 3 = 8 ) represents a parabola intersecting the horizontal line ( y = 8 ). The expanded equation ( x^2 + 2x - 11 = 0 ) finds where this curve crosses ( y = 0 ). Since the roots are irrational, graphing confirms two real, distinct solutions symmetrically placed about ( x = -1 ), the vertex of the parabola.", "## Practical Applications", "Quadratic equations like this appear in physics (projectile motion), engineering (structural design), and economics (profit modeling). Correctly expanding and solving such equations ensures accurate predictions and decisions based on mathematical models.", "## Summary", "Transforming ( x^2 + 2x - 3 = 8 ) to ( x^2 + 2x - 11 = 0 ) simplifies solving by aligning with standard algebraic procedures. While factoring may be limited, applying the quadratic formula efficiently yields precise solutions. Mastering these techniques builds confidence in tackling real-world problems involving quadratic relationships.", "---", "Keywords: ( x^2 + 2x - 3 = 8 \rightarrow x^2 + 2x - 11 = 0 ), quadratic equations, solving quadratics, quadratic formula, factoring tricks, algebra homework help, quadratic root solutions."]
