A 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.
The solutions of x2 − 2x − 3 = 0 are where its parabola meets the x-axis: x = −1 and x = 3. A parabola can cross the axis twice, touch it once, or miss it.
Formal definition and notation
A quadratic equation in one variable x is an equation equivalent to ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0; if a = 0 the equation is linear. Its solutions, also called roots, are the x-intercepts of the parabolay = ax2 + bx + c. They are given by the quadratic formula, and the discriminantb2 − 4ac tells how many real solutions there are: two if it is positive, one if it is zero, and none if it is negative (the solutions are then complex numbers).
the discriminant: its sign tells how many real solutions there are
Key facts
A quadratic equation has at most two solutions, because its highest power of x is 2.
Factoring uses the zero product property: if (x − 2)(x − 5) = 0, then x − 2 = 0 or x − 5 = 0, so x = 2 or x = 5.
x2 = 9 is quadratic too: x = 3 or x = −3. Taking a square root gives a positive and a negative answer.
The solutions of ax2 + bx + c = 0 are where the graph of y = ax2 + bx + c crosses the x-axis.
Check a solution by substituting it: for x = 3 in x2 − 2x − 3 = 0, 9 − 6 − 3 = 0.
The quadratic formula
The quadratic formula
x = −b ± √b2 − 4ac2a
a, b, c
the coefficients of ax2 + bx + c = 0, with signs; the equation must equal 0 first
±
compute once with + and once with − to get both solutions
b2 − 4ac
positive: 2 real solutions; 0: one; negative: none
The formula always works. Factoring is faster when the numbers are friendly, and taking square roots is fastest when there is no x term, as in x2 = 49.
Quadratic Equation examples, step by step
Example 1: Take square roots
Warm-up
Solve 2x2 = 50.
There is no x term, so isolate x2: divide both sides by 2 to get x2 = 25.
Take the square root of both sides, remembering both signs: x = ±√25.
x = 5 or x = −5. Check: 2(25) = 50 for both.
Answer: x = -5 or x = 5.
Example 2: Solve by factoring
Core
Solve x2 + 2x = 15.
Make one side 0: x2 + 2x − 15 = 0.
Find two numbers that multiply to −15 and add to 2: 5 and −3.
Factor: (x + 5)(x − 3) = 0.
Set each factor to 0: x = −5 or x = 3.
Answer: x = -5 or x = 3.
Example 3: Use the quadratic formula
Test-style
Solve 2x2 − 4x − 3 = 0. Give the solutions to the nearest hundredth.
Here a = 2, b = −4, c = −3. The trinomial does not factor nicely.
Discriminant: (−4)2 − 4(2)(−3) = 16 + 24 = 40. It is positive, so there are two solutions.
x = 4 ± √404, and √40 ≈ 6.325.
x ≈ 10.3254 ≈ 2.58 or x ≈ −2.3254 ≈ −0.58.
Answer: x ≈ 2.58 or x ≈ -0.58 (exactly 1 ± √102).
Common mistakes
✗ Mistakex2 + 2x = 15, so x(x + 2) = 15, and x = 15 or x + 2 = 15.
✓ Insteadx2 + 2x − 15 = 0, so (x + 5)(x − 3) = 0 and x = −5 or x = 3.
Make one side 0 before factoring or using the quadratic formula. The zero product property only works when the product equals 0.
✗ Mistakex2 = 25, so x = 5.
✓ Insteadx = 5 or x = −5, since (−5)2 = 25 too.
A positive number has two square roots. Write ± when you take the square root of both sides.
✗ Mistakex2 = 6x. Divide by x: x = 6.
✓ Insteadx2 − 6x = 0, so x(x − 6) = 0 and x = 0 or x = 6.
Dividing by x throws away the solution x = 0. Move everything to one side and factor instead.
✗ MistakeFor x2 − 6x + 5 = 0, the formula starts with −b = −6.
✓ Insteadb = −6, so −b = 6: x = 6 ± √162, giving 5 or 1.
Copy each coefficient with its sign. The opposite of a negative number is positive.
Not to be confused with
Quadratic Equationvs.Quadratic function
A quadratic function, y = x2 − 2x − 3, describes a whole parabola. A quadratic equation sets the expression equal to a number, x2 − 2x − 3 = 0, and asks which x-values make it true.
x2 − 2x − 3 by itself is a polynomial (an expression). Adding = 0 turns it into a quadratic equation you can solve.
Quick check: 3 questions
Question 1 of 3
Which of these is a quadratic equation?
Show answer and explanation
Answer: B.4x2 = 7x can be written 4x2 − 7x = 0, with highest power 2. Choice A is linear, C is cubic, and D has x2 in a denominator, so it is not a polynomial equation.
Question 2 of 3
What are the solutions of (x − 4)(x + 6) = 0?
Show answer and explanation
Answer: A. Set each factor to 0: x − 4 = 0 gives 4, and x + 6 = 0 gives −6. Choice B flips both signs, C multiplies 4 by 6, and D forgets the second factor.
Question 3 of 3
How many real solutions does x2 + 4x + 9 = 0 have?
Show answer and explanation
Answer: C. The discriminant is 42 − 4(1)(9) = 16 − 36 = −20. It is negative, so the parabola never reaches the x-axis and there are no real solutions.
Quadratic Equation: frequently asked questions
How many solutions can a quadratic equation have?
Zero, one or two real solutions, never more. The discriminant b2 − 4ac tells which: positive means two, zero means one (a double root, where the parabola touches the x-axis), and negative means none. With a negative discriminant there are still two complex solutions, which you meet in Algebra 2.
What are the ways to solve a quadratic equation?
Four main methods. Take square roots when there is no x term (x2 = 49). Factor when the trinomial splits nicely. Complete the square to rewrite it as (x − p)2 = q. Use the quadratic formula, which always works. Graphing shows the solutions as x-intercepts but may only give estimates.
Why can a not be 0 in a quadratic equation?
If a = 0, the x2 term disappears and ax2 + bx + c = 0 becomes bx + c = 0, a linear equation with one solution and a straight-line graph. The quadratic formula would also divide by 2a = 0, which is impossible. The x2 term is what makes the equation quadratic.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-REI.B.4, CCSS HSA-REI.B.4.b
Cite this entry
Effortless Math Team. “Quadratic Equation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/quadratic-equation/
Flashcard
TermQuadratic Equationax2 + bx + c = 0
DefinitionAn equation with an x2 term and a ≠ 0. x2 − 5x + 6 = 0 factors to (x − 2)(x − 3) = 0: x = 2 or 3.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2X-InterceptAn x-intercept is a point where a graph crosses or touches the x-axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Algebra · Grade 8, Algebra 1, Algebra 2ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryQuadratic 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 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 2Real SolutionA real solution of an equation is a solution that is a real number, so it lies on the number line, such as x = 3 or x = −√2. Some equations have none: x2 + 4 = 0 has only imaginary solutions. For a quadratic, the discriminant b2 − 4ac tells how many there are: 2, 1 or 0.Algebra · Algebra 1, Algebra 2Complex NumberA complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, with i2 = −1. The real part is a and the imaginary part is b. Examples: 3 + 2i and −4i. 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In a word problem it is usually a fixed amount, such as a one-time fee.Algebra · Grade 6, Grade 7, Algebra 1PowerA 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 7Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, 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 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 8Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8Completing 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, PrecalculusSystem of EquationsA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Algebra · Grade 8, Algebra 1Quadratic FunctionA quadratic function is a function that can be written f(x) = ax2 + bx + c with a ≠ 0. 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