How to find x intercept algebraically?

Understand the Problem

The question is asking how to find the x-intercept of a function algebraically, which typically involves setting the function equal to zero and solving for the variable x.

Answer

The x-intercept of a function is found by solving $f(x) = 0$. For linear functions $x = -\frac{b}{m}$; for quadratic functions $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Steps to Solve

Set the function equal to zero To find the x-intercept of the function, start by setting the function equal to zero. For example, if the function is $f(x) = ax^2 + bx + c$, set it as:

$$ ax^2 + bx + c = 0 $$

Rearrange the equation If necessary, rearrange the equation to isolate terms. If it's a quadratic, it is already in the standard form. If it's a linear function like $f(x) = mx + b$, rearranging will look like:

$$ mx + b = 0 $$

Solve for x Next, solve for $x$. If it's linear, subtract $b$ from both sides and then divide by $m$:

$$ mx = -b \implies x = -\frac{b}{m} $$

If it's quadratic, use the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Check your solutions Finally, substitute the x-values back into the original function to ensure they produce a zero output, confirming they are indeed x-intercepts.

The x-intercept can be found using the following methods:

For a linear function $f(x) = mx + b$, the x-intercept is:

$$ x = -\frac{b}{m} $$

For a quadratic function $f(x) = ax^2 + bx + c$, the x-intercepts are:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

More Information

The x-intercept represents the point where the graph of the function crosses the x-axis. It is a vital concept in graphing functions, as it helps in determining where the function equals zero.

Tips

Forgetting to set the function equal to zero initially.
Misapplying the quadratic formula or forgetting the plus/minus ($\pm$) when multiple solutions exist.
Not simplifying the final answer, leaving it in an incomplete form.

Thank you for voting!