Solving Quadratic Equations: Zero Product and Square Root Properties

Quadratic Equations in One Variable

Definition and Standard Form

A quadratic equation in one variable is an equation that can be written in the standard form:

$ax^2 + bx + c = 0$, where $a \neq 0$

• Quadratic equations are second-degree polynomial equations in one variable.

• The coefficients $a$, $b$, and $c$ are real numbers, with $a$ not equal to zero.

• Examples: $x^2 - 17x = -12$, $2x^2 + 7x = 0$

Zero Product Property

Statement and Application

The Zero Product Property states that if the product of two expressions is zero, then at least one of the expressions must be zero:

If $AB = 0$, then $A = 0$ or $B = 0$

• This property is fundamental in solving quadratic equations that have been factored.

• For example, if $(x-1)(x+7) = 0$, then $x-1 = 0$ or $x+7 = 0$.

• Solving each equation gives $x = 1$ or $x = -7$.

Steps for Solving Quadratic Equations Using the Zero Product Property

1. Write the original equation in standard form: $ax^2 + bx + c = 0$

2. Use the addition property of equality to move all terms to one side.

3. Factor the quadratic expression (using methods such as the AC method).

4. Apply the zero product property to set each factor equal to zero.

5. Solve each resulting linear equation for $x$.

Example:

Solve $6x^2 - 17x = -12$

• Move all terms to one side: $6x^2 - 17x + 12 = 0$

• Factor: $(2x - 3)(3x - 4) = 0$

• Set each factor to zero: $2x - 3 = 0$ or $3x - 4 = 0$

• Solve: $x = \frac{3}{2}$ or $x = \frac{4}{3}$

Square Root Property

Statement and Usage

The Square Root Property is used to solve equations of the form $x^2 = c$:

If $x^2 = c$, then $x = \pm \sqrt{c}$

• This property is useful when the quadratic equation can be rewritten so that the variable term is squared and isolated.

• It is often used after completing the square or when the equation is already in the form $x^2 = c$.

Steps for Solving Quadratic Equations Using the Square Root Property

1. Add or subtract terms to isolate the squared variable term.

2. Divide by the coefficient of the squared term if necessary.

3. Take the square root of both sides, remembering to include both the positive and negative roots.

Examples:

• Example 1: $x^2 - 16 = 0$ Add 16 to both sides: $x^2 = 16$ Take square root: $x = \pm 4$

• Example 2: $2x^2 + 72 = 0$ Subtract 72: $2x^2 = -72$ Divide by 2: $x^2 = -36$ Take square root: $x = \pm 6i$ Additional info: The solution is imaginary because the square root of a negative number is not real.

• Example 3: $(x - 19)^2 = 7$ Take square root: $x - 19 = \pm \sqrt{7}$ Solve for $x$: $x = 19 \pm \sqrt{7}$

Summary Table: Properties for Solving Quadratic Equations

Property	Form of Equation	Solution Method	Example
Zero Product Property	Factored form: $(ax + b)(cx + d) = 0$	Set each factor to zero and solve	$(x - 1)(x + 7) = 0$ $x = 1$ or $x = -7$
Square Root Property	Squared variable: $x^2 = c$	Take square root of both sides	$x^2 = 16$ $x = \pm 4$