How to Find the X Intercept

Two Methods:Simple 2-Variable EquationsFor Quadratic Equations

In algebra, 2-dimensional coordinate graphs have a horizontal axis, or x-axis, and a vertical axis, or y-axis. The places where lines representing a range of values cross these axes are called intercepts. The y-intercept is the place where the line crosses the y-axis and the x-intercept where the line crosses the x-axis. Finding the x-intercept algebraically can be simple or involved, depending on whether the equation is a simple 2-variable or a quadratic equation. The steps below cover how to do so for both kinds of equation.

Method 1
Simple 2-Variable Equations

  1. Image titled Find the X Intercept Step 1
    Substitute 0 for the value of y. At the point the line of values crosses the horizontal axis, the value of y is 0.
    • In the example equation 2x + 3y = 6, substituting 0 for y transforms the equation to 2x + 3(0) = 6, or simply 2x = 6.
  2. Image titled Find the X Intercept Step 2
    Solve for x. This usually means dividing both sides of the equation by the coefficient showing for x to give it a value of 1.
    • In the example equation given above, 2x = 6, dividing both sides by 2 produces 2/2 x = 6/2, or x = 3. This is the x-intercept for the equation 2x + 3y = 6.
    • You can use the same steps for equations in the form ax^2 + by^2 = c. In this case, after substituting 0 for y, you'll be left with x^2 = c/a, and after finding the value to the right of the equal sign, you'll then have to find the square root of x-squared. This will yield 2 values, 1 positive and 1 negative, which, when added together, equal 0.

Method 2
For Quadratic Equations

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    Put the equation in the form ax^2 + bx + c = 0. This is the standard form for writing a quadratic equation, where a represents the coefficient for x-squared, b represents the coefficient for x, and c is a purely numeric value.
    • For the example in this section, we'll use the equation x^2 +3x - 10 = 0.
  2. Image titled Find the X Intercept Step 4
    Solve the equation for x. There are several ways to solve a quadratic equation. The 2 we'll cover here are factoring and using the quadratic formula.
    • Factoring involves breaking a quadratic equation into 2 simpler algebraic expressions that, when multiplied together, produce the quadratic equation. Often the values of a and c can be keys to figuring out the correct factors. Because 2 times 5 equals 10, the absolute value of c, and because the absolute value of b is less than that of c, 2 and 5 are likely the numeric components of the correct factors. Because 5 minus 2 equals 3, the correct factors are x + 5 and x - 2. Substituting the factors for the quadratic equation, (x + 5)(x - 2) = 0, the 2 x-intercepts are -5 (-5 + 5 = 0) and 2 (2 - 2 = 0).
    • Using the quadratic formula involves plugging the values for a, b, and c from the quadratic equation into the formula (-b + or - SQR (b^2 - 4 ac))/2a (where SQR represents the square root) to find the value or values for x.
    • Inserting the values of 1, 3, and -10 into this equation yields (-3 + or - SQR (3^2 - 4(1)(-10)))/2(1). The value inside the SQR parentheses reduces to 9 -(-40) or 9+40, which is 49, so the equation reduces to (-3 + or - 7)/2, which leads to (-3 + 7)/2 or 4/2, which is 2, and (-3 -7)/2 or -10/2, which is -5.
    • Unlike the simple 2-variable equations described in the previous section, quadratic equations are graphed on a coordinate graph in the form of a parabola (a curve resembling a "U" or "V") instead of a straight line. Quadratic equations may have no x-intercept, 1 x-intercept, or 2 x-intercepts.


  • In the example equation given under "For a Simple 2-Variable Equation," if you substituted a value of 0 for x instead of for y, you could then find the value of the y-intercept.

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Categories: Algebra