Solving Quadratic Equations by Completing the Square

Solve quadratic equations using the completing the square method with detailed step-by-step solutions

Quadratic Equation Coefficients

1x² + 6x + 5 = 0

Standard form: ax² + bx + c = 0

Coefficient a

Coefficient of x² (cannot be 0)

Coefficient b

Coefficient of x

Coefficient c

Constant term

Show step-by-step solution

Show complex solutions

Decimal precision

Solution

Solutions

x₁ = -1.0000

x₂ = -5.0000

Two distinct real solutions

Equation Properties

Discriminant (Δ):16.0000

Vertex x-coordinate:-3.0000

Vertex y-coordinate:-4.0000

Parabola opens:Upward

Vertex Form:

y = 1(x - -3.0000)² + -4.0000

Step-by-step Solution

Step 1: Original quadratic equation

1x² +6x +5 = 0

Start with the quadratic equation in standard form ax² + bx + c = 0

Step 2: Move the constant term to the right side

x² +6.0000x = -5.0000

Isolate the terms with x on the left side

Step 3: Complete the square

x² +6.0000x + (3.0000)² = -5.0000 + (3.0000)²

Add (b/2a)² = (3.0000)² = 9.0000 to both sides

Step 4: Factor the perfect square trinomial

(x +3.0000)² = 4.0000

The left side is now a perfect square trinomial

Step 5: Take the square root of both sides

x +3.0000 = ±2.0000

Since the right side is positive, we get two solutions

Step 6: Solve for x

x = -3.0000 ± 2.0000

Subtract the constant term from both sides

Step 7: Final solutions

x₁ = -1.0000, x₂ = -5.0000

The equation has two distinct real solutions

Worked Examples

Example 1: Two Solutions

x² + 6x + 5 = 0

Step 1: Complete the square

x² + 6x + 9 = -5 + 9

Step 2: Factor perfect square

(x + 3)² = 4

Step 3: Take square root

x + 3 = ±2

Solutions: x = -1, x = -5

Example 2: No Real Solutions

x² - 2x + 4 = 0

Step 1: Complete the square

x² - 2x + 1 = -4 + 1

Step 2: Factor perfect square

(x - 1)² = -3

Step 3: Analyze result

Since -3 < 0, no real solutions exist

Complex solutions: x = 1 ± i√3

Completing the Square Method

Step 1: Make Monic

Divide by 'a' if a ≠ 1

Step 2: Move Constant

Move c to right side

Step 3: Complete Square

Add (b/2)² to both sides

Step 4: Factor & Solve

Factor and take square root

Discriminant Analysis

Δ = b² - 4ac

Discriminant formula

•

Δ > 0

Two distinct real solutions

•

Δ = 0

One repeated real solution

•

Δ < 0

No real solutions (complex)

Perfect Square Patterns

(x + a)² = x² + 2ax + a²

(x - a)² = x² - 2ax + a²

To complete the square for x² + bx, add (b/2)²

Understanding Completing the Square

What is Completing the Square?

Completing the square is a method for solving quadratic equations by transforming them into a perfect square trinomial. This method converts the standard form ax² + bx + c = 0 into the form (x + d)² = e, which can be easily solved by taking square roots.

Key Steps

Make the coefficient of x² equal to 1 (if necessary)
Move the constant term to the right side
Add (b/2)² to both sides to complete the square
Factor the perfect square trinomial
Take the square root of both sides and solve for x

Why Use This Method?

Completing the square is particularly useful because it:

Always works for any quadratic equation
Provides insight into the vertex form of parabolas
Helps understand the geometric meaning of solutions
Is the foundation for deriving the quadratic formula
Useful for optimization problems

Vertex Form Connection:

Completing the square transforms ax² + bx + c into a(x - h)² + k, where (h, k) is the vertex of the parabola.

Applications

Optimization

Finding maximum or minimum values of quadratic functions

Graphing

Converting to vertex form for easy graphing of parabolas

Engineering

Projectile motion, circuit analysis, and structural optimization

Related Quadratic Calculators

Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula

Completing the Square Calculator

Transform quadratics to vertex form

Factoring Calculator

Factor quadratic expressions and polynomials