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How do you find the intercepts of a quadratic equation?

Published in Quadratic Intercepts 2 mins read

To find the intercepts of a quadratic equation, you need to determine the points where the graph of the equation crosses the x-axis (x-intercepts) and the y-axis (y-intercept).

Finding the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0.

  • Method: Substitute x = 0 into the quadratic equation. The resulting value of y is the y-coordinate of the y-intercept.

    • For example, as shown in the provided video reference, if the quadratic equation is in the form y = ax2 + bx + c, then the y-intercept occurs when x = 0. Therefore, y = a(0)2 + b(0) + c, which simplifies to y = c. So, the y-intercept is at the point (0, c). The video example shows this, indicating that the y-intercept is (0, 4).

Finding the X-Intercepts

The x-intercepts are the points where the graph crosses the x-axis. This occurs when y = 0.

  • Method: Set the quadratic equation equal to 0 and solve for x. This typically involves factoring, using the quadratic formula, or completing the square.

Methods to Solve for X-Intercepts

  1. Factoring: If the quadratic can be factored easily into the form (ax + b)(cx + d) = 0, set each factor equal to 0 and solve for x. This gives you the two x-intercepts.

    • Example: For x2 - 5x + 6 = 0, factoring gives (x - 2)(x - 3) = 0. Setting each factor to 0 results in x=2 and x=3, providing intercepts at (2, 0) and (3, 0)

  2. Quadratic Formula: If factoring is difficult or impossible, use the quadratic formula:

    • For a quadratic in the form ax2 + bx + c = 0, the formula is:

      • x = (-b ± √(b2 - 4ac)) / 2a

    • Applying this formula, calculate two possible values for x, which gives the x-intercepts.

  3. Completing the Square: Convert the quadratic into vertex form, a(x-h)2 + k=0, and then isolate (x-h)2, take the square root, and solve for x

    • This method is particularly helpful to find the vertex, as well.

Summary Table

Intercept Condition Method
Y-intercept x = 0 Substitute x=0 into the equation and solve for y
X-intercepts y = 0 Set the equation equal to 0 and solve for x using factoring, the quadratic formula, or completing the square

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