Quadratic equations are fundamental in algebra, often encountered in various forms. One common representation is the standard form and another useful form is the vertex form. Converting from standard form to vertex form allows for easier identification of the vertex and provides a clearer understanding of the graph’s shape and direction.

In this article, we’ll explore what these forms are, the methods to convert between them, and why vertex form can be particularly advantageous in solving and graphing quadratic equations.

What is Quadratic Equation?

Quadratic Equation is a type of polynomial equation of degree two, which means the highest exponent of the variable is two. Quadratic equations are used to model various real-world situations, such as projectile motion, area problems, and optimization scenarios.

The solutions to a quadratic equation, known as the roots, can be found using various methods, including factoring, completing the square, and the quadratic formula.

Quadratic equations can have different forms such as:

• Standard Form
• Vertex Form

Standard Form of a Quadratic Equation

Standard form of a quadratic equation is typically written as:

ax² + bx + c = 0

Where:

• x is the variable,
• a, b, and c are constants with a ≠ 0.

Components of the Standard Form

• a: coefficient of x². It determines the direction and width of the parabola.
  • If a is positive, the parabola opens upwards; if a is negative, the parabola opens downwards.
• b: coefficient of x. It affects the position and direction of the parabola.
• c: The constant term. It represents the y-intercept of the parabola, which is the point where the graph intersects the y-axis.

Examples of Standard Form

Some examples of quadratic equation in standard form

• 2x² − 4x + 1 = 0

In this equation a = 2, b = −4, and c = 1.

• 3x² + 5x − 2 = 0

In this equation a = 3, b = 5, and c = -2.

• x² – 4x + 4 = 0

In this equation a = 1, b = -4, and c = 4.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is a way to express the equation such that it highlights the vertex of the parabola. The vertex form is written as:

y = a(x − h)² + k

Where:

• (h, k) is the vertex of the parabola,
• a is the same coefficient as in the standard form ax² + bx + c, which affects the width and direction of the parabola.

Components of the Vertex Form

• a: Determines the direction and the width of the parabola.
  • If a is positive, the parabola opens upwards;
  • If a is negative, the parabola opens downwards.
• h: The x-coordinate of the vertex. It represents the horizontal shift of the parabola from the origin.
• k: The y-coordinate of the vertex. It represents the vertical shift of the parabola from the origin.

Examples of Vertex Form

Some examples of quadratic equations in vertex form are:

• y = 2(x − 1)² + 3

In this equation a = 2, h = 1, k = 3, and vertex is (1, 3).

• y = -3(x + 4)² – 5

In this equation a = -3, h = -4, k = -5, and vertex is (-4, -5).

• y = (1/2)(x − 2)² + 7

In this equation a = 1/2, h = 2, k = 7, and vertex is (2, 7).

Conversion from Standard Form to Vertex Form

Converting a quadratic equation from standard form ax² + bx + c to vertex form a(x − h)² + k involves completing the square. Here’s a step-by-step guide to the conversion process:

Step 1: Start with the standard form.

y = ax² + bx + c

Step 2: Factor out the coefficient of x² from the x-terms:

If a ≠ 1, factor out a from the x² and x terms: y = a[x² + (b/a)x] + c

Step 3: Complete the square:

Take the coefficient of x, divide it by 2, and square it: [Tex]\left(\frac{b}{2a}\right)^2[/Tex].

Add and subtract this square inside the parentheses: [Tex]y = a \left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 – \left(\frac{b}{2a}\right)^2 \right) + c[/Tex]

Simplify inside the parentheses to form a perfect square trinomial: [Tex]y = a \left( \left( x + \frac{b}{2a} \right)^2 – \left(\frac{b}{2a}\right)^2 \right) + c[/Tex]

Step 4: Simplify the equation:

Distribute a and simplify the constant terms:

[Tex]\left( x + \frac{b}{2a} \right)^2 – a \left(\frac{b}{2a}\right)^2 + c[/Tex]

⇒ [Tex]y = a \left( x + \frac{b}{2a} \right)^2 – \frac{b^2}{4a} + c[/Tex]

Step 4: Combine the constants:

Combine the constants to get the final vertex form: [Tex]y = a \left( x + \frac{b}{2a} \right)^2 + \left( c – \frac{b^2}{4a} \right)[/Tex]

In the vertex form [Tex]y = a(x-h)^2 + k[/Tex], the vertex (h, k) can be identified as:

• [Tex]h = -\frac{b}{2a}[/Tex]
• [Tex]k = c – \frac{b^2}{4a}[/Tex]

Read More about Completing the Square Method.

Let’s consider the example for better understanding.

Example: Convert y = 2x² + 8x + 5 to vertex form.

Solution:

Step 1: Start with the standard form:

y = 2x² + 8x + 5

Step 2: Factor out the coefficient of x² from the x-terms:

y = 2(x² + 4x) + 5

Step 3: Complete the square:

Take the coefficient of x, divide it by 2, and square it: (4/2)² = 4.

Add and subtract this square inside the parentheses:

y = 2 [x² + 4x + 4 – 4] + 5

⇒ y = 2[(x + 2)² – 4] + 5

Step 4: Simplify the equation:

Distribute 2 and simplify the constants:

y = 2(x + 2)² – 8 + 5

⇒ y = 2(x + 2)² – 3

So, the vertex form of y = 2x² + 8x + 5 is:

y = 2(x + 2)² – 3

The vertex of the parabola is (-2, -3).

Read More,

• Polynomials
• How to find the Discriminant of a Quadratic Equation
• Real-Life Applications of Quadratic Equations
• Standard Equation of a Parabola
• Solving Quadratic Equations by Factoring
• Graphing Quadratic Functions 

Solved Examples of Converting Standard Form to Vertex Form

Example 1: Convert the quadratic equation y = -3x² + 6x – 1 from standard form to vertex form.

Solution:

Given: y = -3x² + 6x – 1

Factor out the coefficient of x² from the first two terms

y = -3(x² – 2x) – 1

Take the coefficient of x, which is -2, divide it by 2, and square it: (-2/2)²

Add and subtract this square inside the parentheses:

y = -3(x² – 2x + 1 – 1) – 1

⇒ y = -3((x – 1)² – 1) – 1

⇒ y = -3(x – 1)² + 3 – 1

⇒ y = -3(x – 1)² + 2

So, the vertex form of y = -3x² + 6x – 1 is:

y = -3(x – 1)² + 2

Example 2: Convert the quadratic equation y = x² – 4x + 7 from standard form to vertex form.

Solution:

Given: y = x² – 4x + 7

Factor out the coefficient of x² from the first two terms (coefficient is 1 here, so no factoring needed)

y = x² – 4x + 7

Take the coefficient of x, which is -4, divide it by 2, and square it: (-4/2)² = 4

Add and subtract this square inside the parentheses:

y = x² – 4x + 4 – 4 + 7

⇒ y = (x – 2)² – 4 + 7

⇒ y = (x – 2)² + 3

So, the vertex form of y = x² – 4x + 7 is:

y = (x – 2)² + 3

The vertex is (2, 3).

Example 3: Convert the quadratic equation y = 2x² – 8x – 5 from standard form to vertex form.

Solution:

Given: y = 2x² – 8x – 5

Factor out the coefficient of x² from the first two terms

y = 2(x² – 4x) – 5

Take the coefficient of x, which is 4, divide it by 2, and square it: (4/2)² = 4

Add and subtract this square inside the parentheses:

y = 2(x² + 4x + 4 – 4) – 5

y = 2((x – 2)² – 4) – 5

Distribute the 2 and combine like terms:

y = 2(x – 2)² – 8 – 5

y = 2(x – 2)² – 13

So, the vertex form of y = 2x² + 8x + 5 is:

y = 2(x + 2)² – 3

Practice Problems on Standard Form to Vertex Form

Problem 1: Convert the quadratic equation y = 3x² + 12x + 5 from standard form to vertex form.

Problem 2: Convert the quadratic equation y = -2x² + 8x – 6 from standard form to vertex form.

Problem 3: Convert the quadratic equation y = x² – 6x + 10 from standard form to vertex form.

Problem 4: Convert the quadratic equation y = -4x² + 16x – 7 from standard form to vertex form.

Problem 5: Convert the quadratic equation y = 5x² + 20x + 15 from standard form to vertex form.

FAQs on Standard Form to Vertex Form

Define quadratic equation.

A quadratic equation is a type of polynomial equation of the second degree, typically in the form:

ax² + bx + c = 0

where:

• a, b, and c are constants with a ≠ 0,
• x represents an unknown variable.

What is the standard form of a quadratic equation?

The standard form of a quadratic equation is: y = ax² + bx + c where a, b, and c are constants.

What is the vertex form of a quadratic equation?

The vertex form of a quadratic equation is: y = a(x − h)² + k where (h, k) is the vertex of the parabola.

What does “completing the square” mean?

Completing the square is a method used to form a perfect square trinomial from a quadratic expression.