Solving Equations of the Form x^2 = p
Solving Equations of the Form x² = p introduces Grade 6 students to extracting square roots to solve quadratic-type equations, a concept in Yoshiwara Elementary Algebra Chapter 5: Exponents and Roots. When x² = k (with k > 0), there are two solutions: x = √k and x = -√k, since both positive and negative values square to the same result. This skill is the gateway to understanding quadratic equations and irrational numbers.
Key Concepts
Property Taking a square root is the opposite of squaring a number. To solve an equation of the form $x^2 = k$ (where $k 0$), we take the square root of both sides. Because a positive number has two square roots, the solution is written as:.
$$x = \pm\sqrt{k}$$.
Examples To solve the equation $x^2 = 81$, we take the square root of both sides. The solutions are $x = \pm\sqrt{81}$, which means $x = 9$ and $x = 9$.
Common Questions
How do you solve x squared equals a number?
Take the square root of both sides. Remember that x² = k gives two answers: x = √k and x = -√k, because both values square to k.
Why are there two solutions when solving x² = p?
Because both a positive and a negative number can produce the same square. For example, both 4 and -4 satisfy x² = 16.
What if k is negative in x² = k?
If k is negative, there are no real solutions because squaring any real number always gives a non-negative result.
Where is solving x² = p covered in Yoshiwara Elementary Algebra?
This topic is covered in Chapter 5: Exponents and Roots of Yoshiwara Elementary Algebra.
What is the connection between x² = p and quadratic equations?
Solving x² = p is the simplest case of a quadratic equation with no linear term. It builds the foundation for solving more complex quadratics using the quadratic formula.