What are the roots of the equation? $m^2+10=26$
The given problem asks us to find the roots of the quadratic equation \(m^2 + 10 = 26\). Our goal is to isolate the variable \(m\) and identify which of the provided options—\(\pm 4\), \(\pm 8\), \(\pm 2\), or \(\pm 16\)—satisfies the equation. A quadratic equation of this form, \(m^2 = k\), can be solved by taking the square root of both sides, remembering to account for both the positive and negative roots.
To solve \(m^2 + 10 = 26\), we first isolate the term containing \(m\). We subtract 10 from both sides of the equation:
\[m^2 = 26 - 10\]
\[m^2 = 16\]
Now that we have \(m^2 = 16\), we can find the value of \(m\) by taking the square root of both sides:
\[m = \pm \sqrt{16}\]
Since the square root of 16 is 4, we have:
\[m = \pm 4\]
We have determined that the roots of the equation \(m^2 + 10 = 26\) are \(m = 4\) and \(m = -4\). Comparing this result with the given choices, we see that it corresponds to option (a). Therefore, the correct roots are \(\pm 4\).
In real life, quadratic equations are essential for calculating dimensions and movement. For instance, if you are designing a square garden and you know its area must be 16 square meters, you need to find the side length \(s\). Using the formula for the area of a square, \(s^2 = 16\), you solve for \(s\) and find that the side must be 4 meters. This simple principle is the foundation for various architectural and engineering tasks, where we often need to determine physical dimensions from squared constraints.