The standard form of a quadratic is ax2 + bx + c, with the terms in descending powers of x and a ≠ 0. As an equation it is ax2 + bx + c = 0; as a function, y = ax2 + bx + c. It shows a, b and c directly, which the quadratic formula needs, and c is the y-intercept.
In Spanish
forma estándar de una cuadrática
Also called
standard form, general form of a quadratic, ax^2 + bx + c form
All three forms describe one parabola. Standard form is the one with a, b and c in plain view.
Formal definition and notation
A quadratic expression in x is in standard form when written ax2 + bx + c, where a, b, c are real constants and a ≠ 0, with like terms combined and powers in decreasing order. A quadratic equation is in standard form when written ax2 + bx + c = 0. Some textbooks call this the general form and use standard form for the vertex forma(x − h)2 + k.
Vertex form shows the vertex and factored form shows the zeros; standard form shows the y-intercept and the coefficients.
Every quadratic can be written in standard form, but not every quadratic has a real factored form.
Standard form and what it gives you
Standard form and what it gives you
y = ax2 + bx + c, a ≠ 0
direction
opens up if a > 0, down if a < 0
y-intercept
(0, c)
axis of symmetry
x = −b2a
solutions of = 0
x = −b ± √b2 − 4ac2a
To convert another form to standard form, expand (FOIL or square the binomial) and combine like terms.
Standard Form of a Quadratic examples, step by step
Example 1: Rearrange into standard form
Warm-up
Write 7 − 3x + 2x2 = 4 in standard form and name a, b, c.
Subtract 4 so one side is 0: 3 − 3x + 2x2 = 0.
Order by descending power: 2x2 − 3x + 3 = 0.
Read the coefficients with signs.
Answer:2x2 − 3x + 3 = 0, so a = 2, b = -3, c = 3.
Example 2: From vertex form
Core
Write y = −2(x + 3)2 + 5 in standard form.
Square the binomial: (x + 3)2 = x2 + 6x + 9.
Multiply by −2: −2x2 − 12x − 18.
Add 5: y = −2x2 − 12x − 13.
Answer:y = −2x2 − 12x − 13, so a = -2, b = -12, c = -13.
Example 3: From the zeros
Test-style
A quadratic has zeros x = −1 and x = 6 and leading coefficient 3. What is its constant term c in standard form?
Factored form: y = 3(x + 1)(x − 6).
FOIL: (x + 1)(x − 6) = x2 − 5x − 6.
Multiply by 3: y = 3x2 − 15x − 18.
Answer: c = -18 (the standard form is 3x2 − 15x − 18).
Common mistakes
✗ MistakeIn x2 − 7x + 10, b = 7.
✓ Insteadb = −7: the sign in front of a term belongs to its coefficient.
Using b = 7 in −b2a or the quadratic formula flips the answers.
✗ Mistake5x − x2 + 2 is already standard form, so a = 5.
✓ InsteadReorder: −x2 + 5x + 2, so a = −1, b = 5, c = 2.
a is the coefficient of x2, not the first number you see. Sort by power first.
✗ Mistakex2 + 4x = 12 has c = 12.
✓ InsteadMake one side 0: x2 + 4x − 12 = 0, so c = −12.
Standard form for an equation needs 0 on one side. Moving 12 across changes its sign.
Not to be confused with
Standard Form of a Quadraticvs.Vertex form
Vertex form a(x − h)2 + k shows the vertex (h, k). Some college textbooks call vertex form the standard form and ax2 + bx + c the general form, so check which one your course means.
For a line, standard form is Ax + By = C. For a quadratic it is ax2 + bx + c, with an x2 term and descending powers.
Quick check: 3 questions
Question 1 of 3
What is b in 4 − x2 + 9x = 0 after writing it in standard form?
Show answer and explanation
Answer: C. Standard form is −x2 + 9x + 4 = 0, so b = 9. 4 is c, −1 is a, and −9 flips the sign without reason.
Question 2 of 3
Which is y = (x − 4)2 − 3 in standard form?
Show answer and explanation
Answer: A.(x − 4)2 = x2 − 8x + 16, and 16 − 3 = 13. The second choice skipped the middle term, the third subtracted 3 from −16, and the fourth used −4x instead of −8x.
Question 3 of 3
Which equation is NOT quadratic once written in standard form?
Show answer and explanation
Answer: C. Subtracting x2 from both sides leaves 3x + 6 = 0, so a = 0 and the equation is linear. The other three keep an x2 term after simplifying.
Standard Form of a Quadratic: frequently asked questions
What is the standard form of a quadratic equation?
It is ax2 + bx + c = 0, where a, b and c are real numbers and a is not 0. All terms are on one side in order of decreasing powers, and the other side is 0. For example, x2 = 6x − 5 becomes x2 − 6x + 5 = 0, with a = 1, b = −6 and c = 5.
Why is standard form useful?
It puts the coefficients in plain view. You need a, b and c for the quadratic formula, the discriminant and the axis of symmetry x = −b2a. The constant c is the y-intercept, and the sign of a tells which way the parabola opens. It is also the usual starting point for factoring.
How do you convert vertex form to standard form?
Expand and simplify. Square the binomial, multiply by a, and add k. For y = 2(x − 1)2 + 3: (x − 1)2 = x2 − 2x + 1, times 2 gives 2x2 − 4x + 2, and adding 3 gives y = 2x2 − 4x + 5. Going the other way uses completing the square.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-SSE.B.3.a, CCSS HSA-SSE.B.3.b, CCSS HSF-IF.C.8.a
Cite this entry
Effortless Math Team. “Standard Form of a Quadratic.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/standard-form-of-a-quadratic/
Flashcard
TermStandard Form of a Quadraticax2 + bx + c = 0
DefinitionTerms in descending powers, one side 0, a ≠ 0. x2 = 6x − 5 becomes x2 − 6x + 5 = 0.
Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2Vertex FormVertex form is a way to write a quadratic function as y = a(x − h)2 + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)2 + 1 has its vertex at (3, 1).Algebra · Algebra 1, Algebra 2Leading CoefficientThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x2 − 5x4 + 2x, the highest power is x4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph’s ends point.Algebra · Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8Constant TermA constant term is a term in an expression or equation that has no variable, so its value never changes. In 3x2 − 5x + 7, the constant term is 7. Its sign belongs to it: in x2 + 4x − 9 it is −9. In a word problem it is usually a fixed amount, such as a one-time fee.Algebra · Grade 6, Grade 7, Algebra 1Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax2 + bx + c = 0 with a ≠ 0: x = −b ± √b2 − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Standard Form of a Linear EquationThe standard form of a linear equation is Ax + By = C, where A, B and C are integers and A and B are not both 0. Both variables sit on the left side. It makes the intercepts easy to find: the x-intercept is CA, the y-intercept is CB, and the slope is −AB.Algebra · Grade 8, Algebra 1PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2DiscriminantThe discriminant of a quadratic equation ax2 + bx + c = 0 is the number b2 − 4ac. Its sign tells how many real solutions the equation has without solving it: positive means two, zero means one (a repeated root), and negative means none, because the parabola never touches the x-axis.Algebra · Algebra 1, Algebra 2Completing the SquareCompleting the square is a method that rewrites x2 + bx as a perfect square by adding (b2)2: x2 + bx + (b2)2 = (x + b2)2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra · Algebra 1, Algebra 2, PrecalculusQuadratic FunctionA quadratic function is a function that can be written f(x) = ax2 + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola, which opens up when a > 0 and down when a < 0. Its highest or lowest point is the vertex.Functions · Algebra 1, Algebra 2, Precalculus