acalculator

What is the x-intercept of my graph?

Pick how your equation is written and type its numbers. The x intercept calculator shows where the graph crosses the x-axis, as exact fractions for lines, and where it crosses the y-axis.

Your numbers

Equation
x-intercept
(4, 0)

The x-intercept of y = 2x - 8 is (4, 0).

x-intercept x
4
y-intercept
(0, -8)
Equation
y = 2x - 8

x-intercept: (4, 0). The x-intercept of y = 2x - 8 is (4, 0).

How to calculate

Finds the x-intercept (and the y-intercept) of a line in slope-intercept form, through two points, or in standard form, or of a parabola y = ax² + bx + c.

Example with the default inputs (Equation y = mx + b, Slope (m) 2, y-intercept (b) -8): The x-intercept of y = 2x - 8 is (4, 0).

Method: Set y = 0 and solve for x: y = mx + b gives x = −b ÷ m; Ax + By = C gives x = C ÷ A; two points give m = (y₂ − y₁) ÷ (x₂ − x₁) first; y = ax² + bx + c gives x = (−b ± √(b² − 4ac)) ÷ 2a.

  • Coefficients are read as the exact decimals typed; line intercepts are exact fractions.
  • A parabola’s irrational x-intercepts are 64-bit floats, shown to 10 significant figures in the points.
  • With a = 0, y = ax² + bx + c is the line y = bx + c.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Equation y = mx + b, Slope (m) 0.5, y-intercept (b) -3 gives x-intercept (6, 0), x₁ 6, y-intercept (0, -3), Equation y = 1/2 x - 3.Source: OpenStax, Algebra and Trigonometry 2e, section 4.1 Linear Functions (x-intercept where f(x) = 0, y-intercept b), https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02, example: f(x) = ½x − 3 crosses the x-axis at (6, 0)
  2. Equation y = ax² + bx + c, a 3, b 5, c -2 gives x-intercept (-2, 0) and (1/3, 0), x₁ -2, x₂ 0.333333, Discriminant (b² − 4ac) 49.Source: OpenStax, Algebra and Trigonometry 2e, section 5.1 Quadratic Functions (x-intercepts from the quadratic formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions, retrieved 2026-10-02, example 7: 3x² + 5x − 2 = (3x − 1)(x + 2), x-intercepts (−2, 0) and (1/3, 0)
  3. Equation y = ax² + bx + c, a 2, b 4, c -4 gives x₁ -2.732051, x₂ 0.732051, x-intercept (-2.732050808, 0) and (0.7320508076, 0).Source: OpenStax, Algebra and Trigonometry 2e, section 5.1 Quadratic Functions (x-intercepts from the quadratic formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions, retrieved 2026-10-02, example 8: 2x² + 4x − 4 has x-intercepts −1 ± √3, about (−2.732, 0) and (0.732, 0)
  4. Equation y = mx + b, Slope (m) 2, y-intercept (b) -8 gives x-intercept (4, 0), x₁ 4, y-intercept (0, -8), Equation y = 2x - 8.Source: OpenStax, Algebra and Trigonometry 2e, section 4.1 Linear Functions (x-intercept where f(x) = 0, y-intercept b), https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02
  5. Equation Two points, x₁ 1, y₁ -2, x₂ 4, y₂ 4 gives x-intercept (2, 0), x₁ 2, y-intercept (0, -4), Equation y = 2x - 4.Source: OpenStax, Algebra and Trigonometry 2e, section 4.1 Linear Functions (x-intercept where f(x) = 0, y-intercept b), https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02
  6. Equation Ax + By = C, A 3, B 4, C 12 gives x-intercept (4, 0), y-intercept (0, 3), Equation y = -3/4 x + 3.Source: OpenStax, Algebra and Trigonometry 2e, section 4.1 Linear Functions (x-intercept where f(x) = 0, y-intercept b), https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-1-linear-functions, retrieved 2026-10-02
  7. Equation Ax + By = C, A 3, B 0, C 8 gives x-intercept (8/3, 0), x₁ 2.666667, y-intercept None: the line does not cross the y-axis, Equation x = 8/3.
  8. Equation y = ax² + bx + c, a 1, b -2, c -3 gives x-intercept (-1, 0) and (3, 0), x₁ -1, x₂ 3, Discriminant (b² − 4ac) 16, y-intercept (0, -3), Equation y = x² - 2x - 3.Source: OpenStax, Algebra and Trigonometry 2e, section 5.1 Quadratic Functions (x-intercepts from the quadratic formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions, retrieved 2026-10-02
  9. Equation y = ax² + bx + c, a 1, b 0, c -2 gives x-intercept (-1.414213562, 0) and (1.414213562, 0), x₁ -1.414214, x₂ 1.414214, Discriminant (b² − 4ac) 8.Source: OpenStax, Algebra and Trigonometry 2e, section 5.1 Quadratic Functions (x-intercepts from the quadratic formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions, retrieved 2026-10-02
  10. Equation y = ax² + bx + c, a 1, b 2, c 5 gives x-intercept None: the graph does not cross the x-axis, Discriminant (b² − 4ac) -16.Source: OpenStax, Algebra and Trigonometry 2e, section 5.1 Quadratic Functions (x-intercepts from the quadratic formula), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions, retrieved 2026-10-02
  11. Equation y = mx + b, Slope (m) 0, y-intercept (b) 5 gives x-intercept None: the graph does not cross the x-axis, y-intercept (0, 5), Equation y = 5.

How it works

Every number is from −10^9 to 10^9 and is read as the exact decimal typed (0.1 is exactly one tenth). The x-intercepts are where y = 0; the y-intercept is where x = 0.

y = mx + b. If m ≠ 0, the x-intercept is x = −b ÷ m. If m = 0 and b ≠ 0, there is none; if m = 0 and b = 0, the line is the x-axis. The y-intercept is (0, b).

Two points (x₁, y₁) and (x₂, y₂). The same two points give no answer: "The two points are the same, so they do not fix one line." If x₁ = x₂, the line is the vertical line x = x₁: its x-intercept is (x₁, 0), and it has no y-intercept unless x₁ = 0, when it is the y-axis. Otherwise m = (y₂ − y₁) ÷ (x₂ − x₁) and b = y₁ − m × x₁, and the rules for y = mx + b apply.

Ax + By = C. A = B = 0 gives no answer: "A and B cannot both be 0: that is not a line." If B = 0, the line is the vertical line x = C ÷ A (as for two points). Otherwise m = −A ÷ B and b = C ÷ B.

y = ax² + bx + c. If a = 0, it is the line y = bx + c. Otherwise the discriminant is D = b² − 4ac, worked out exactly:

  • D < 0: no x-intercept.
  • D = 0: one, x = −b ÷ 2a.
  • D > 0 and D the square of a fraction: two exact roots, (−b − √D) ÷ 2a and (−b + √D) ÷ 2a.
  • D > 0 otherwise: two roots as 64-bit floats. With p = b ÷ a, r = c ÷ a and E = D ÷ a² worked exactly and then taken as the nearest floats, t = −(p + s√E) ÷ 2, with s = 1 for p ≥ 0 and −1 for p < 0; the roots are t and r ÷ t.

When a root or the discriminant is beyond the 64-bit float range (very large and very small coefficients together), there is no answer: "The numbers are too far apart in size to show the answer."

The y-intercept of a parabola is (0, c).

The calculator shows:

  • x-intercept: each point as (x, 0), smaller x first, joined by and. Exact values are fractions in lowest terms (8/3, -2, 1/3, no mixed numbers); float roots are written to 10 significant figures in their shortest form (1.414213562). With none: None: the graph does not cross the x-axis; for the x-axis itself: Every point: the graph is the x-axis.
  • x-intercept x and Second x-intercept x: the x values as numbers (the second only for a parabola with two).
  • y-intercept: (0, y) with y as a fraction in lowest terms; None: the line does not cross the y-axis for a vertical line x = c with c ≠ 0, or Every point: the line is the y-axis for x = 0.
  • Equation: a line as y = mx + b with fractions in lowest terms (y = 2x - 8, y = -x, y = -3/4 x + 3 with a space after a fractional slope, y = 5 when m = 0, y = 2x when b = 0) or x = c for a vertical line; a parabola as typed, zero terms left out and a coefficient of 1 or −1 written as a sign only (y = x² - 2x - 3), with numbers to 10 significant figures.
  • Discriminant (b² − 4ac): for a parabola with a ≠ 0.

Worked examples by hand

y = ½x − 3. 0 = ½x − 3, so x = 6: (6, 0); y-intercept (0, −3); equation y = 1/2 x - 3.

y = 2x − 8. x = 8 ÷ 2 = 4: (4, 0); y-intercept (0, −8).

Through (1, −2) and (4, 4). m = (4 − (−2)) ÷ (4 − 1) = 2, b = −2 − 2 × 1 = −4, so y = 2x - 4, x = 4 ÷ 2 = 2: (2, 0); y-intercept (0, −4).

3x + 4y = 12. y = 0 gives x = 12 ÷ 3 = 4; x = 0 gives y = 12 ÷ 4 = 3; m = −3/4, so y = -3/4 x + 3.

3x + 0y = 8. The vertical line x = 8/3 (2.6667): x-intercept (8/3, 0), no y-intercept.

y = x² − 2x − 3. D = 4 + 12 = 16 = 4², so x = (2 ∓ 4) ÷ 2: (−1, 0) and (3, 0); y-intercept (0, −3).

y = 3x² + 5x − 2. D = 25 + 24 = 49 = 7², x = (−5 ∓ 7) ÷ 6: (−2, 0) and (1/3, 0).

y = 2x² + 4x − 4. D = 16 + 32 = 48, not a square: x = −1 ± √3 = −2.732050808 and 0.7320508076.

y = x² − 2. D = 8: x = ±√2 = ±1.414213562.

y = x² + 2x + 5. D = 4 − 20 = −16: no x-intercept.

y = 5 (m = 0, b = 5). A horizontal line above the axis: no x-intercept; y-intercept (0, 5).

Other questions people ask

What is an x-intercept?

The point where a graph crosses the x-axis. There y = 0, so the point is (x, 0). For f(x) = ½x − 3, setting 0 = ½x − 3 gives x = 6, so the x-intercept is (6, 0).

How do I find the x-intercept of y = mx + b?

Set y = 0 and solve: x = −b ÷ m. For y = 2x − 8, x = 8 ÷ 2 = 4. A horizontal line (m = 0) has no x-intercept, unless it is the x-axis itself.

How do I find the x-intercept from two points?

Find the slope m = (y₂ − y₁) ÷ (x₂ − x₁), then b = y₁ − m × x₁, then x = −b ÷ m. Through (1, −2) and (4, 4): m = 6 ÷ 3 = 2, b = −4, x = 2.

How do I find the intercepts of Ax + By = C?

Set y = 0 to get x = C ÷ A, and set x = 0 to get y = C ÷ B. For 3x + 4y = 12, the x-intercept is (4, 0) and the y-intercept is (0, 3).

How many x-intercepts can a parabola have?

Two, one, or none, depending on the discriminant b² − 4ac. Above 0: two, from x = (−b ± √(b² − 4ac)) ÷ 2a. Equal to 0: one, where the vertex touches the axis. Below 0: none.

What is the difference between an x-intercept, a zero and a root?

They name the same number from different views. A zero of f is an x where f(x) = 0; a root is a solution of f(x) = 0; the x-intercept is the point (x, 0) on the graph.

What is the y-intercept?

Where the graph crosses the y-axis, at x = 0. For y = mx + b it is (0, b); for y = ax² + bx + c it is (0, c). A vertical line x = c has none, unless c = 0.