Solving Quadratic Equations: Factoring, Square Roots, and Completing the Square

Quadratic Equations

A quadratic equation in x is an equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$ and $a, b, c$ are real numbers. Quadratic equations are also called second-degree polynomial equations in x.

1. Solving Quadratic Equations by Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic as a product of two binomials and applying the Zero Product Principle.

• Zero Product Principle: If $AB = 0$, then $A = 0$ or $B = 0$.

• Steps:

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

  2. Factor the quadratic expression completely.

  3. Set each factor equal to zero and solve for x.

• Example 1: $x^2 - 5x + 6 = 0$

  • Factor: $(x-2)(x-3) = 0$

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

  • Solutions: $x=2$, $x=3$

• Example 2: $4x^2 - 9x = 0$

  • Factor: $x(4x-9) = 0$

  • Set each factor to zero: $x=0$ or $4x-9=0$

  • Solutions: $x=0$, $x=\frac{9}{4}$

2. Solving Quadratic Equations by the Square Root Property

This method is used when the quadratic equation can be written in the form $(x-a)^2 = d$.

• Square Root Property: If $u^2 = d$, then $u = \sqrt{d}$ or $u = -\sqrt{d}$.

• If $d$ is negative, introduce the imaginary unit $i$ where $i = \sqrt{-1}$.

• Steps:

  1. Isolate the squared term.

  2. Take the square root of both sides, remembering to consider both the positive and negative roots.

  3. Solve for x.

• Example 1: $x^2 = 25$

  • Take square root: $x = \pm 5$

• Example 2: $x^2 = -9$

  • Take square root: $x = \pm \sqrt{-9} = \pm 3i$

• Example 3: $(x-2)^2 = 6$

  • Take square root: $x-2 = \pm \sqrt{6}$

  • Solutions: $x = 2 \pm \sqrt{6}$

3. Solving Quadratic Equations by Completing the Square

Completing the square is a method that rewrites a quadratic equation in the form $(x-h)^2 = k$ to solve for x. This method is especially useful when the quadratic cannot be easily factored.

• Perfect Square Trinomials:

  • $(A+B)^2 = A^2 + 2AB + B^2$

  • $(A-B)^2 = A^2 - 2AB + B^2$

• Steps:

  1. If necessary, rewrite the equation in standard form $ax^2 + bx + c = 0$.

  2. Divide both sides by $a$ (if $a \neq 1$) to make the coefficient of $x^2$ equal to 1.

  3. Move the constant term to the other side of the equation.

  4. Add the square of half the coefficient of $x$ to both sides to complete the square.

  5. Rewrite the left side as a squared binomial.

  6. Apply the square root property and solve for x.

• Example: $x^2 + 6x + 5 = 0$

  • Move constant: $x^2 + 6x = -5$

  • Take half of 6, square it: $(6/2)^2 = 9$

  • Add 9 to both sides: $x^2 + 6x + 9 = 4$

  • Rewrite: $(x+3)^2 = 4$

  • Take square root: $x+3 = \pm 2$

  • Solutions: $x = -3 \pm 2$ so $x = -1$ or $x = -5$

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Key Steps
Factoring	Quadratic is factorable over integers	Rewrite in standard form, factor, set each factor to zero
Square Root Property	No linear term ($bx$), or equation is a perfect square	Isolate squared term, take square root of both sides
Completing the Square	Quadratic not easily factorable	Rewrite, complete the square, solve using square root property

Note: For all methods, always check your solutions by substituting back into the original equation.

Additional info: The quadratic formula is another universal method for solving any quadratic equation, but it is not covered in detail in these notes.