Use the zero product property to find the solutions to the equation $x^2-13 x+30=0$.
To solve the quadratic equation \(x^2 - 13x + 30 = 0\), we look for two numbers whose sum is the coefficient of the middle term (\(-13\)) and whose product is the constant term (\(30\)). Let these numbers be \(p\) and \(q\). We require:
\[p + q = -13\]
\[p \cdot q = 30\]
By examining the factors of \(30\), we identify that \(-3\) and \(-10\) satisfy both conditions:
\[(-3) + (-10) = -13\]
\[(-3) \cdot (-10) = 30\]
Thus, the quadratic expression can be factored as \((x - 3)(x - 10)\).
Now that we have the equation in the factored form \((x - 3)(x - 10) = 0\), we apply the zero-product property. This property states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore:
\[x - 3 = 0 \quad \text{or} \quad x - 10 = 0\]
Solving these linear equations individually:
So, the solutions to the equation are \(x = 3\) and \(x = 10\). Comparing this result with the given options, this corresponds to option c.
Quadratic equations like this are frequently used in modeling physical phenomena, such as predicting the trajectory of a projectile. If you were to launch a ball and its height \(h\) at time \(t\) were given by a quadratic expression, finding the roots of that equation would help you determine the times when the ball hits the ground. Learning to factor these equations efficiently allows you to solve such real-world problems quickly and accurately.