Extraction of roots
Extraction of Roots is a method for solving equations that contain a squared expression by isolating it and taking the square root of both sides, including both positive and negative roots. Covered in Yoshiwara Elementary Algebra Chapter 9: More About Exponents and Roots, this Grade 6 algebra technique is especially useful for solving equations of the form (ax + b)² = c. Students must remember to apply ± when extracting the root, yielding two solutions whenever c > 0.
Key Concepts
Property To solve an equation by extraction of roots, first isolate the squared expression. Then, take the square root of both sides, remembering to include both the positive and negative roots ($\pm$). Finally, solve the resulting linear equations.
Examples Solve $(x+3)^2 = 49$. Take the square root of both sides: $x+3 = \pm\sqrt{49} = \pm 7$. This gives two equations: $x+3=7$ (so $x=4$) and $x+3= 7$ (so $x= 10$).
Solve $3(y 1)^2 = 75$. First, isolate the squared part by dividing by 3: $(y 1)^2 = 25$. Take the square root: $y 1 = \pm 5$. The two solutions are $y = 1+5=6$ and $y=1 5= 4$.
Common Questions
What is extraction of roots in algebra?
Extraction of roots is a method to solve equations by isolating a squared expression and taking the square root of both sides, always writing ± to account for both positive and negative roots.
How do you solve an equation by extraction of roots?
Isolate the squared term, take the square root of both sides, include the ± symbol, then solve the resulting two linear equations.
Why must you use ± when extracting square roots?
Because both a positive and negative number can produce the same square. Using ± ensures you find all solutions.
Where is extraction of roots in Yoshiwara Elementary Algebra?
This method is taught in Chapter 9: More About Exponents and Roots of Yoshiwara Elementary Algebra.
What is the difference between extraction of roots and the quadratic formula?
Extraction of roots works directly when the squared expression is isolated and there is no linear term. The quadratic formula works for any quadratic equation.