["Solving Equation: ( x^2 + (x + 1)^2 = 145 )", "Finding and solving quadratic equations is a fundamental skill in algebra, widely used in math, physics, engineering, and data science. One such common problem is the equation:", "[
\nx^2 + (x + 1)^2 = 145
\n]", "This article guides you step-by-step through solving this equation, explains its real-world applications, and highlights essential algebraic techniques.", "---", "### What Is the Equation ( x^2 + (x+1)^2 = 145 )?", "The equation combines two quadratic expressions: ( x^2 ) and ( (x + 1)^2 ). By expanding and simplifying, it transforms into a standard quadratic form that can be solved using algebraic methods.", "---", "### Step-by-Step Solution", "Step 1: Expand the equation
\nExpand ( (x+1)^2 ):", "[
\n(x+1)^2 = x^2 + 2x + 1
\n]", "Now substitute back into the original equation:", "[
\nx^2 + (x^2 + 2x + 1) = 145
\n]", "Step 2: Combine like terms", "[
\nx^2 + x^2 + 2x + 1 = 145
\n]
\n[
\n2x^2 + 2x + 1 = 145
\n]", "Step 3: Move all terms to one side", "[
\n2x^2 + 2x + 1 - 145 = 0
\n]
\n[
\n2x^2 + 2x - 144 = 0
\n]", "Step 4: Simplify the quadratic equation", "Divide every term by 2:", "[
\nx^2 + x - 72 = 0
\n]", "---", "### Step 5: Solve the simplified quadratic equation", "We now solve ( x^2 + x - 72 = 0 ) using:", "- Factoring:
\nWe look for two numbers that multiply to (-72) and add to (1). Those numbers are (9) and (-8):", "[
\n(x + 9)(x - 8) = 0
\n]", "Set each factor equal to zero:", "[
\nx + 9 = 0 \quad \Rightarrow \quad x = -9
\n]
\n[
\nx - 8 = 0 \quad \Rightarrow \quad x = 8
\n]", "- Verification using quadratic formula (optional):
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \quad \ ext{where } a=1, b=1, c=-72
\n]
\n[
\nx = \frac{-1 \pm \sqrt{1^2 - 4(1)(-72)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 288}}{2} = \frac{-1 \pm \sqrt{289}}{2} = \frac{-1 \pm 17}{2}
\n]
\n[
\nx = \frac{16}{2} = 8 \quad \ ext{or} \quad x = \frac{-18}{2} = -9
\n]", "Both methods yield the same solutions.", "---", "### Solutions to the Equation", "[
\n\boxed{x = 8 \quad \ ext{or} \quad x = -9}
\n]", "---", "### Real-World Applications", "Equations like ( x^2 + (x+1)^2 = 145 ) model real-life scenarios involving displacement, distance, height differences, or optimization problems. For example:
\n- Projectile motion: Calculating time or position at certain intervals.
\n- Geometry: Finding positions on a line where squared distances add to a known value.
\n- Financial modeling: Balancing two investment outcomes whose returns follow simple quadratic forms.", "---", "### Key Takeaways", "- Expanding binomial squares is essential when dealing with expressions like ( (x+1)^2 ).
\n- Simplifying quadratic equations makes them easier to solve via factoring or the quadratic formula.
\n- Verifying solutions ensures no extraneous roots are accepted.
\n- This type of problem enhances algebraic fluency, crucial in STEM disciplines.", "---", "If you're studying algebra or preparing for competitive exams, mastering equations of this form helps build a strong foundation for more complex mathematical modeling.", "---", "Keywords: ( x^2 + (x+1)^2 = 145 ), quadratic equation, solve ( x^2 + (x+1)^2 ), algebraic methods, factoring, real-world applications, quadratic formula, step-by-step solution, math tutoring tips."]
