A 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.
Each crossing of the x-axis is one real solution. The sign of the discriminant tells you which picture you have without graphing.
Formal definition and notation
A real solution of an equation f(x) = 0 is a value x ∈ ℝ that makes it true; graphically, it is the x-coordinate of a point where the graph of y = f(x) meets the x-axis. For ax2 + bx + c = 0 with real coefficients and a ≠ 0, the discriminantD = b2 − 4ac decides: D > 0 gives two distinct real solutions, D = 0 one repeated real solution, and D < 0 no real solutions (two non-real complex solutions).
An imaginary number like 5i can be a solution, but not a real one. When D < 0 the quadratic solutions are non-real, of the form a ± bi.
Real Solutionvs.Rational solution
A real solution can be irrational: x2 = 2 has the real solutions ±√2. Real only means it lies on the number line.
Quick check: 3 questions
Question 1 of 3
How many real solutions does x2 − 4x + 4 = 0 have?
Show answer and explanation
Answer: C.D = (−4)2 − 4(1)(4) = 16 − 16 = 0, so there is one real solution, x = 2. 2 would need D > 0, and 0 would need D < 0.
Question 2 of 3
Which equation has no real solutions?
Show answer and explanation
Answer: D. For D, 12 − 4(1)(1) = −3 < 0. A has ± 3, B factors as (x + 3)(x − 1), and C has the single solution 0.
Question 3 of 3
The graph of y = x2 − 2x − 8 crosses the x-axis at how many points?
Show answer and explanation
Answer: A.D = (−2)2 − 4(1)(−8) = 4 + 32 = 36 > 0, so two real solutions and two x-intercepts: x = 4 and x = −2. A parabola never crosses an axis 3 times.
Real Solution: frequently asked questions
What does no real solution mean?
It means no real number makes the equation true. On a graph, the curve never meets the x-axis. The equation may still have solutions that are complex numbers. For example, x2 + 1 = 0 has no real solution, but i and −i both work because i2 = −1.
How do you know how many real solutions a quadratic has?
Write it as ax2 + bx + c = 0 and compute the discriminant b2 − 4ac. Positive means two real solutions, zero means one, and negative means none. You can also graph y = ax2 + bx + c and count the x-intercepts. Both methods always agree.
Can a real solution be a fraction or an irrational number?
Yes. Real numbers include integers, fractions and irrational numbers, so x = 23 and x = √5 are real solutions. The word real only rules out non-real complex numbers such as 2 + 3i. Many quadratics have irrational real solutions that come out of the quadratic formula.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-REI.B.4.b
Cite this entry
Effortless Math Team. “Real Solution.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/real-solution/
Flashcard
TermReal Solutionb2 − 4ac
DefinitionA solution that is a real number, an x-intercept of the graph. x2 = 9 has two (± 3); x2 = −9 has none.
X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7CoefficientA 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 8DiscriminantThe 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 2Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6X-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 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. Complex numbers let you take square roots of negatives, such as √−9 = 3i.Number System · Algebra 2, PrecalculusSquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3RootsThe 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 2Real 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 1ParabolaA 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, GeometrySquare 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 1SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8Imaginary NumberAn imaginary number is a real number multiplied by i, the imaginary unit, where i = √−1 and i2 = −1. Examples are 3i, −12i and √−25 = 5i. Imaginary numbers give square roots of negative numbers a value, so equations like x2 = −9 have solutions: x = ± 3i.Number System · Algebra 2, PrecalculusIrrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · 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 2Quadratic 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 2VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6