9.6 Solve Equations with Square Roots

Solving equations with square roots is a crucial skill in algebra. It involves isolating radical terms, squaring both sides, and carefully verifying solutions. This process helps us tackle real-world problems involving areas, gravity, and speed.

Square root equations can be tricky due to extraneous solutions. We must always check our answers in the original equation to ensure accuracy. This topic builds on our understanding of functions, inverses, and domains in algebra.

Solving Equations with Square Roots

Radical equation solving techniques

• Isolate the radical term on one side of the equation
  • Perform identical operations on both sides to preserve equality (addition, subtraction, multiplication, division)
  • Manipulate the equation to get the radical term alone on one side
• Square both sides of the equation to remove the radical
  • Apply the property $\sqrt{x}^2 = x$ to eliminate the square root
  • Be aware that squaring both sides may introduce extraneous solutions (solutions that satisfy the simplified equation but not the original)
• Solve the resulting equation after squaring
  • Simplify the squared terms by expanding and combining like terms
  • Isolate the variable using inverse operations (addition, subtraction, multiplication, division)
• Verify the solutions by substituting them into the original radical equation
  • Confirm that each solution satisfies the original equation before squaring
  • Check if the solution is within the domain of the original equation

Square root formulas in applications

• Area formulas involving square roots
  • Triangle area: $A = \frac{1}{2}bh$, where $b$ is the base length and $h$ is the height
  • Circle area: $A = \pi r^2$, where $r$ is the radius
  • Pythagorean theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are leg lengths and $c$ is the hypotenuse length (right triangles)
• Gravity formula: $d = \frac{1}{2}gt^2$
  • $d$ represents the distance an object falls
  • $g$ is the acceleration due to gravity (9.8 m/s² on Earth)
  • $t$ is the time in seconds the object is falling
• Speed formula: $v = \sqrt{2ad}$
  • $v$ represents the speed of an object
  • $a$ is the acceleration the object experiences
  • $d$ is the distance the object travels

Extraneous solutions in radical equations

• Extraneous solutions arise when squaring both sides of an equation
  • They satisfy the simplified squared equation but not the original radical equation
• Identifying extraneous solutions:
  1. Substitute each solution into the original radical equation
  2. Check if the solution satisfies the original equation before squaring
• Eliminating extraneous solutions:
  • Discard any solutions that do not satisfy the original radical equation
  • Include only the solutions that work in the original equation in the final solution set

Related concepts

• Functions and their inverses:
  • A function is a relation where each input has exactly one output
  • The inverse function reverses the operation of the original function
  • Square root functions are inverse functions of quadratic equations
• Domain and range:
  • The domain is the set of all possible input values for a function
  • The range is the set of all possible output values for a function
  • For square root functions, the domain is typically restricted to non-negative numbers