Solving Quadratic Equations by Difference of Squares
Solving quadratic equations by difference of squares in Algebra 1 (California Reveal Math, Grade 9) uses the factoring pattern a² - b² = (a - b)(a + b). Set each factor equal to zero: a - b = 0 gives a = b, and a + b = 0 gives a = -b. For example, x² - 9 = 0 factors to (x - 3)(x + 3) = 0, giving x = 3 or x = -3. The two solutions are always equal and opposite. Recognizing the difference of squares pattern is a fast, elegant method that avoids the quadratic formula for these specific cases.
Key Concepts
Property To solve a quadratic equation in the form of a difference of squares, factor the expression and apply the Zero Product Property: $$a^2 b^2 = 0 \implies (a b)(a + b) = 0$$ Then, set each factor to zero: $a b = 0$ or $a + b = 0$.
Examples Solve $x^2 36 = 0$: Factor to $(x 6)(x + 6) = 0$. Setting each factor to zero gives $x = 6$ or $x = 6$. Solve $4x^2 49 = 0$: Factor to $(2x 7)(2x + 7) = 0$. Setting each factor to zero gives $2x = 7 \implies x = \frac{7}{2}$ and $2x = 7 \implies x = \frac{7}{2}$. Solve $25x^2 1 = 0$: Factor to $(5x 1)(5x + 1) = 0$. Setting each factor to zero gives $x = \frac{1}{5}$ or $x = \frac{1}{5}$.
Explanation A difference of squares occurs when a quadratic equation consists of two perfect squares separated by a subtraction sign. To solve these equations, you first factor the expression into two binomials representing the sum and difference of the square roots. After factoring, you apply the Zero Product Property by setting each binomial equal to zero to find the solutions.
Common Questions
What is the difference of squares factoring pattern?
a² - b² = (a - b)(a + b). This factors any expression that is a perfect square minus another perfect square.
How do you solve a quadratic equation using difference of squares?
Factor the expression as (a - b)(a + b), then apply the Zero Product Property: set each factor equal to zero and solve. The two solutions are a = b and a = -b.
Can you show an example?
4x² - 25 = 0: rewrite as (2x)² - 5² = 0, factor as (2x - 5)(2x + 5) = 0. Solutions: x = 5/2 and x = -5/2.
How do you recognize when to use the difference of squares method?
Look for two perfect square terms with a minus sign between them and no middle term. Both terms must be perfect squares (x², 4, 9, 25, 100, etc.).
Where is solving by difference of squares covered in California Reveal Math Algebra 1?
This technique is taught in California Reveal Math, Algebra 1, as part of Grade 9 quadratic equations and polynomial factoring.
What if the expression is a sum of squares like x² + 9?
A sum of squares (a² + b²) does not factor over the real numbers. This is different from the difference of squares and has no real factored form.
How are the two solutions always related in difference of squares?
The two solutions are always equal and opposite: a = b and a = -b. They are additive inverses of each other.