Which graph can be used to find the solution(s) to $x^2-1=3$?

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02-11-2026
Mathematics

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Which graph can be used to find the solution(s) to $x^2-1=3$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T21:24:45-05:00

To solve $x^2 - 1 = 3$, you can find the intersection points of the parabola $y = x^2 - 1$ and the line $y = 3$.
Alternatively, you can rearrange the equation to $x^2 - 4 = 0$ and find the x-intercepts of the parabola $y = x^2 - 4$.
Both methods yield the same solutions: $\boxed{x = 2, x = -2}$.

Explanation

Problem Analysis

To find the solution to the equation $x^2 - 1 = 3$, we are looking for values of $x$ that satisfy the equality. We can approach this graphically in two primary ways, each providing a visual way to identify the roots of the equation. Analysis of the equation shows that it involves a quadratic term and constants, which can be expressed as a function or a system of functions.

Method 1: Intersection of Two Graphs

The first method is to treat the equation as a system of two functions. We set each side of the equation equal to $y$:
\[y_1 = x^2 - 1\]
\[y_2 = 3\]
The solutions to the equation $x^2 - 1 = 3$ are the x-coordinates of the intersection points of the parabola $y = x^2 - 1$ and the horizontal line $y = 3$. At these intersection points, both functions yield the same y-value, confirming that the equality $x^2 - 1 = 3$ holds.

Method 2: Zeros of a Single Graph

The second method is to rearrange the equation to equal zero:
\[x^2 - 1 - 3 = 0\]
\[x^2 - 4 = 0\]
Now, we define a single function $f(x) = x^2 - 4$. The solutions to the original equation are the values of $x$ for which $f(x) = 0$. Graphically, these are the x-intercepts (or roots) of the parabola $y = x^2 - 4$. At these points, the graph crosses the x-axis, where $y = 0$. Both methods are mathematically equivalent and valid for finding the solutions.

Conclusion

In conclusion, there are two primary graphical representations to find the solution to $x^2 - 1 = 3$:

The intersection of the graphs of $y = x^2 - 1$ and $y = 3$.
The x-intercepts of the graph of the function $y = x^2 - 4$.

Either of these approaches provides the visual solution to the equation. Thus, the solutions are found at the x-coordinates where the graphs intersect or touch the axis, which are $\boxed{x = 2, x = -2}$.

Examples

In real-world applications, such as physics or engineering, you might use these graphical methods to find the time at which an object reaches a specific height. For example, if the height of a projectile is given by $h(t) = t^2 - 1$, you can find the time $t$ it hits the 3-meter mark by observing where the graph of $h(t)$ intersects the horizontal line $h = 3$. This visualization helps in quickly estimating results and understanding the behavior of dynamic systems before performing precise calculations.