The equation $x^2-1 x-90=0$ has solutions $\{a, b\}$. What is $a+b$ ?
The given quadratic equation is \(x^2 - x - 90 = 0\). This is in the standard form \(Ax^2 + Bx + C = 0\), where \(A = 1\), \(B = -1\), and \(C = -90\). We are asked to find the sum of the roots \(a + b\), where \(a\) and \(b\) are the solutions to this equation.
According to Vieta's formulas, for a quadratic equation \(Ax^2 + Bx + C = 0\) with roots \(a\) and \(b\), the sum of the roots is given by \(a + b = -\frac{B}{A}\). Substituting the values we identified, \(A = 1\) and \(B = -1\), we calculate:
\[a + b = -\frac{-1}{1} = 1.\]
Thus, the sum of the roots is \(1\).
We have determined that the sum of the roots \(a + b\) is \(1\). Comparing this result with the provided options, we see that the correct choice is \(1\).
In real-life scenarios, quadratic equations are used to model various phenomena, such as the path of a projectile. For example, if you throw a ball, its height over time might be modeled by a quadratic equation. Knowing the properties of the roots, like their sum or product, can quickly provide useful information about the system's behavior without needing to find the individual roots. In this case, Vieta's formulas offer an efficient way to understand the relationship between the equation's coefficients and its roots.