Standard Form Calculator — Linear Equations, Polynomials & Numbers (Free)
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A standard form calculator rewrites a line, a polynomial or a number in its standard form. A linear equation becomes Ax + By = C, so y = (2/3)x − 4 becomes 2x − 3y = 12. A polynomial is written from the highest power down: (x + 3)(x − 1) = x² + 2x − 3. A number becomes a × 10ⁿ: 87,000 = 8.7 × 10⁴.
Want the line as y = mx + b instead? The Slope Intercept Form Calculator rewrites Ax + By = C, two points or point-slope form in slope-intercept form, with steps and a graph.
In standard form, y = (2/3)x − 4 is 2x − 3y = 12 (A = 2, B = −3, C = 12).
Show steps
- Start with the slope-intercept form y = (2/3)x − 4.
- Subtract (2/3)x from both sides so the x-term moves to the left.
- Write the equation with the x- and y-terms on the left and the constant on the right: −(2/3)x + y = −4.
- Clear the fractions: multiply every term by 3, the least common denominator: −2x + 3y = −12.
- Standard form has a positive leading coefficient, so multiply both sides by −1: 2x − 3y = 12.
- A, B and C are integers with no common factor (other than 1), so the equation is in standard form.
- Read the slope as −A/B = 2/3 and the y-intercept as C/B = −4.
Standard form rules used here (common in US algebra courses): A, B and C are integers, A is not negative, and A, B and C have no common factor. Some books also allow fractions or a negative A.
Which standard form do you need?
“Standard form” has different meanings in different classes and countries. The calculator has one tab for each common meaning:
| Where you see it | Standard form means | Example |
|---|---|---|
| Algebra 1, linear equations (US) | Ax + By = C with integer A, B, C | y = 2x + 3 → 2x − y = −3 |
| Polynomials and quadratics | Like terms combined, highest power first | x² = 5x − 6 → x² − 5x + 6 = 0 |
| UK maths, GCSE, science | a × 10ⁿ, with a from 1 up to (not including) 10 | 0.00052 → 5.2 × 10⁻⁴ |
The UK “standard form” of a number is what US classes call scientific notation. In US elementary school, “standard form” of a number simply means writing it with digits (three hundred forty-five is 345). For E notation, engineering notation and arithmetic with powers of ten, use the scientific notation converter.
How to use the standard form calculator
- Pick a tab: Linear equation, Polynomial or Number.
- For a line, choose what you start from: slope-intercept form, standard form (to convert back), point-slope form, two points, or any linear equation you type.
- Type your values. Fractions (2/3), decimals (−0.5) and brackets all work.
- Read the standard form, the other form of the answer, the slope and intercepts or the degree and coefficients, and open Show steps for the full working.
Standard form of a linear equation: Ax + By = C
A linear equation is in standard form when the x- and y-terms are on the left, the constant is on the right, and A, B and C are integers. Most US textbooks also ask that A is not negative and that A, B and C have no common factor. Under those rules each line has exactly one standard form, which makes answers easy to compare.
Convert slope-intercept form to standard form
- Move the x-term to the left side: y = (2/3)x − 4 becomes −(2/3)x + y = −4.
- Clear fractions by multiplying every term by the least common denominator, here 3: −2x + 3y = −12.
- If A is negative, multiply by −1: 2x − 3y = 12.
- Divide by any common factor of A, B and C (here there is none).
Decimal slopes work the same way. y = −0.5x + 4 becomes 0.5x + y = 4, and multiplying by 2 gives x + 2y = 8. Its slope is −1/2, its y-intercept is 4 and its x-intercept is 8.
Convert standard form back to slope-intercept form
Solve for y. For 3x − 2y = 8, subtract 3x to get −2y = −3x + 8, then divide by −2: y = (3/2)x − 4, which is y = 1.5x − 4. In general, the slope is −A/B, the y-intercept is C/B and the x-intercept is C/A.
Vertical and horizontal lines
If B = 0 the line is vertical, x = C/A, and it has no slope-intercept form. If A = 0 the line is horizontal, y = C/B, with slope 0. For example, 0x + 3y = 9 simplifies to y = 3.
Standard form of a polynomial
A polynomial is in standard form when its like terms are combined and its terms are written in descending order of degree. The highest power is the degree, and its coefficient is the leading coefficient.
- (x + 3)(x − 1) = x² − x + 3x − 3 = x² + 2x − 3, so a = 1, b = 2 and c = −3.
- 2x(x − 3) + 5 − x³ = 2x² − 6x + 5 − x³ = −x³ + 2x² − 6x + 5, a cubic with leading coefficient −1.
- y = 2(x − 1)² + 3 is in vertex form. Expanding gives y = 2x² − 4x + 5.
A quadratic equation is in standard form when it reads ax² + bx + c = 0. Move every term to one side and combine: x² = 5x − 6 becomes x² − 5x + 6 = 0, ready for factoring or the quadratic formula.
Standard form of a number: a × 10ⁿ
Write the number as a × 10ⁿ, where a is at least 1 and less than 10 and n is a whole number. Move the decimal point until exactly one non-zero digit is in front of it, and count the moves. Moving left gives a positive n (large numbers); moving right gives a negative n (small numbers).
| Ordinary number | Standard form | Decimal point moves |
|---|---|---|
| 87,000 | 8.7 × 10⁴ | 4 places left |
| 3,450,000 | 3.45 × 10⁶ | 6 places left |
| 459,608 | 4.59608 × 10⁵ | 5 places left |
| 0.00052 | 5.2 × 10⁻⁴ | 4 places right |
| 0.000381 | 3.81 × 10⁻⁴ | 4 places right |
A number like 12 × 10³ is not in standard form because 12 is bigger than 10. Rewrite 12 as 1.2 × 10¹ and add the powers: 1.2 × 10⁴.
Common mistakes
- Leaving fractions in Ax + By = C. (1/2)x + y = 4 is not finished. Multiply by 2: x + 2y = 8.
- Forgetting a common factor. 4x + 6y = 12 simplifies to 2x + 3y = 6.
- Sign slips when moving terms. A term changes sign when it moves across the equals sign.
- Ordering a polynomial by coefficient size. The order is by power of x, not by how big the numbers are.
- Getting the sign of n backwards. Small numbers (below 1) have a negative power of ten.
Standard form practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Subtract 3x: −3x + y = −5. Multiply by −1 so A is positive: 3x − y = 5.
Move the x-term: −(1/2)x + y = 3. Multiply by −2 to clear the fraction and make A positive: x − 2y = −6.
A, B and C must be integers with no common factor and A positive. The others are the same line but have a common factor, a negative A or a decimal.
Move the decimal point 4 places left to get 7.2, so the power is 10⁴. 72 × 10³ has the right value but 72 is not between 1 and 10.
Move the decimal point 3 places right to reach 6.3, so the power is −3.
Quiz complete
Frequently asked questions
How do you convert slope-intercept form to standard form?
Move the x-term to the left, clear fractions, make A positive and divide by any common factor. For y = (3/4)x − 2: −(3/4)x + y = −2, multiply by 4 to get −3x + 4y = −8, then by −1: 3x − 4y = 8.
How do you convert standard form to slope-intercept form?
Solve for y. For 5x + 2y = 10, subtract 5x to get 2y = −5x + 10, then divide by 2: y = −2.5x + 5. The slope is −A/B = −5/2.
What is standard form for a number?
In UK maths and in science, it is a × 10ⁿ with a at least 1 and less than 10, the same as scientific notation. For example, 3,450,000 = 3.45 × 10⁶ and 0.00052 = 5.2 × 10⁻⁴.
How do you put standard form in a calculator?
Type the number a, press the EXP, EE or ×10ˣ key, then type the power. For 3.2 × 10⁵, type 3.2, EXP, 5. Many calculators then show it as 3.2E5. This calculator also accepts 3.2 × 10^5 and 3.2e5.
Does standard form have to be simplified?
For linear equations, usually yes: A, B and C should be integers with no common factor, and A should not be negative. So 4x + 6y = 12 is written 2x + 3y = 6. Some teachers accept any integer form, so check your course rules.
What is the standard form of a quadratic?
A quadratic in standard form is ax² + bx + c (or ax² + bx + c = 0 for an equation). Expand and combine first: (x + 3)(x − 1) = x² + 2x − 3, so a = 1, b = 2 and c = −3. The form a(x − h)² + k is called vertex form.
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