Factored Form of an Equation
Factored form of an equation is a Grade 7 math concept from Yoshiwara Intermediate Algebra where a quadratic or polynomial is expressed as a product of its factors. This form makes it easy to identify roots and solve equations by applying the zero-product property.
Key Concepts
Property The solutions of the quadratic equation $a(x r 1)(x r 2) = 0$ are $r 1$ and $r 2$. This is called the factored form of the quadratic equation. If you know the two solutions of a quadratic equation, you can work backwards to reconstruct the equation.
Examples A quadratic equation has solutions $x=3$ and $x= 6$. The factors are $(x 3)$ and $(x ( 6))$, or $(x+6)$. The equation is $(x 3)(x+6)=0$, which expands to $x^2+3x 18=0$.
To find an equation with solutions $x=2$ and $x=\frac{1}{4}$, start with factors $(x 2)$ and $(x \frac{1}{4})$. For integer coefficients, use $(x 2)$ and $(4x 1)$. The equation is $(x 2)(4x 1)=0$, or $4x^2 9x+2=0$.
Common Questions
What is the factored form of a quadratic equation?
The factored form of a quadratic is written as a(x - r1)(x - r2) = 0, where r1 and r2 are the roots. For example, x^2 - 5x + 6 in factored form is (x - 2)(x - 3).
How do you solve an equation in factored form?
Set each factor equal to zero using the zero-product property. If (x - 2)(x - 3) = 0, then x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
Why is factored form useful in algebra?
Factored form immediately reveals the roots or x-intercepts of the equation, making it the most efficient form for solving quadratic and polynomial equations.
How do you convert standard form to factored form?
Factor the quadratic expression by finding two numbers that multiply to the constant term and add to the coefficient of the middle term, then write it as a product of two binomials.