Quadratic Standard Form Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadratic Standard Form.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The standard form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a \neq 0$ and $a$ , $b$ , $c$ are real number coefficients.

Think of it as a template with three slots: $a$ controls the width and direction of the parabola, $b$ shifts it sideways, and $c$ slides it up or down. Every quadratic can be written this way by expanding and collecting like terms.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Standard form lines a quadratic up as $ax^2+bx+c=0$ so its coefficients are ready to read.

Common stuck point: The procedure for quadratic standard form is the easy part; the trap is forgetting $a\ne0$ . Asking "Is this a single-variable degree-2 equation written as everything-minus-everything $=0$ with $a\ne0$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this a single-variable degree-2 equation written as everything-minus-everything $=0$ with $a\ne0$ ?

Worked Examples

Example 1

easy

Write

3 - 5x + 2x^2

in standard form and identify

a

b

c

Answer

2x^2 - 5x + 3; \quad a=2, b=-5, c=3

First step

Standard form

ax^2 + bx + c

requires terms in decreasing order of exponent.

Full solution

2
Rearrange the expression by decreasing power: $2x^2 - 5x + 3$ .
3
Identify the coefficients: $a = 2$ , $b = -5$ , $c = 3$ .

Standard form

ax^2 + bx + c

always has the

x^2

term first, the

x

term second, and the constant last. The leading coefficient

a

must be nonzero.

Example 2

medium

Convert

y = -(x-2)^2 + 5

to standard form.

Example 3

easy

Expand and write

(x-3)(x+2) = 0

in standard form.

Example 4

medium

Convert

y = (x+3)^2 - 4

to standard form.

Example 5

medium

Write the equation of a parabola with

x

-intercepts

2

and

-5

and leading coefficient

1

in standard form.

Example 6

medium

Convert

y = -2(x-3)^2 + 5

to standard form.

Example 7

medium

Find the

x

-coordinate of the vertex of

y = 2x^2 - 12x + 7

using standard form.

Example 8

hard

Given the parabola

y = ax^2 + bx + c

passes through

(0, 3)

(1, 0)

, and

(-1, 8)

, find

a, b, c

Example 9

hard

Convert

y = 3(2x+1)^2 - 4

to standard form.

Example 10

hard

Convert

y = -(x-2)(x+6)

to standard form and identify the

y

-intercept.

Example 11

challenge

A parabola in standard form

y = ax^2 + bx + c

has vertex

(-1, -8)

and passes through

(2, 10)

. Find

a, b, c

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

4x - x^2 + 1 = 0

in standard form? If not, rewrite it.

Example 2

medium

Convert

(x+1)(x-3) = 0

to standard form.

Example 3

easy

Identify

a

b

c

3x^2 + 5x + 2 = 0

Example 4

easy

Identify

a

-x^2 + 4x - 7 = 0

Example 5

easy

5x + 2 = 0

a quadratic equation?

Example 6

easy

Write

x^2 = 9

in standard form.

Example 7

easy

Identify

b

2x^2 - 7x + 3 = 0

Example 8

easy

Write

3x^2 + 2x = 5

in standard form.

Example 9

easy

x^2 + 3x - 10 = 0

, what is

c

Example 10

easy

Expand and write in standard form:

(x+1)(x+4) = 0

Example 11

medium

Write

2(x-1)^2 = 6

in standard form

ax^2+bx+c=0

Example 12

medium

Write

\frac{x^2}{2} + x = 3

in standard form with integer coefficients.

Example 13

medium

Combine and write in standard form:

x^2 + 3x = 2x^2 - x + 4

Example 14

medium

Identify

a

b

c

7 - 2x + x^2 = 0

(reorder first).

Example 15

medium

Write the quadratic with

a=2

b=-3

c=1

in standard form.

Example 16

medium

A rectangle has length

x+2

and width

x

, area

15

. Write the standard-form equation.

Example 17

medium

Write

(2x-1)(x+3) = 0

in standard form.

Example 18

medium

Write

5x = 2x^2 - 3

in standard form with positive leading coefficient.

Example 19

medium

Expand

3(x-2)(x+1)

and write in standard form.

Example 20

challenge

A quadratic in standard form has

a=1

, sum of roots

5

, product of roots

6

. Write it.

Example 21

challenge

For what

a

does

ax^2 + 6x + 3 = 0

become linear instead of quadratic?

Example 22

challenge

Two quadratics

x^2 + bx + c = 0

share roots

2

and

-3

. Find

b

and

c

Example 23

easy

Write

7x - 2x^2 + 5 = 0

in standard form.

Example 24

easy

Write

5x^2 = 3x + 2

in standard form.

Example 25

easy

Identify

c

x^2 - 6x = 0

Example 26

medium

Convert

y = 2(x-1)(x+4)

to standard form.

Example 27

medium

In standard form, what does the constant term

c

represent on the graph?

Example 28

medium

For

2x^2 + 3x - 5 = 0

, compute the discriminant

b^2 - 4ac

Example 29

medium

Convert

(2x-1)(x+3) = 0

to standard form.

Example 30

hard

A quadratic in standard form has

c = -12

and roots

3

and

-4

. Find the leading coefficient

a

Example 31

hard

For what value of

k

kx^2 - 4x + 1 = 0

not a quadratic equation?

Example 32

hard

Write a quadratic in standard form whose roots are

\tfrac{1}{2}

and

-3

with integer coefficients.

← Back to Quadratic Standard Form Formula Explained Practice Mode

Related Concepts

Quadratic Functions Expressions Quadratic Vertex Form Quadratic Factored Form Completing the Square Discriminant

Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionsexpressions