Use the zero product property to find the solutions to the equation $x^2+12=7 x$.
To solve the quadratic equation \(x^2 + 12 = 7x\), we first rewrite it in standard form by moving all terms to one side. Subtracting \(7x\) from both sides, we obtain:
\[x^2 - 7x + 12 = 0\]
This is a standard quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a=1, b=-7,\) and \(c=12\).
To solve the equation \(x^2 - 7x + 12 = 0\), we look to factor the quadratic expression \(x^2 - 7x + 12\). We need two numbers that multiply to \(12\) and add up to \(-7\). By inspection, these numbers are \(-3\) and \(-4\). Thus, we can write the quadratic as:
\[(x - 3)(x - 4) = 0\]
The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two linear equations:
\(x - 3 = 0\) or \(x - 4 = 0\)
Solving these equations individually:
Thus, the solutions to the original equation are \(x = 3\) and \(x = 4\), which corresponds to option (a).
The zero product property is incredibly useful in real-world scenarios where you need to find critical points or solve for unknowns. For example, if you are modeling the flight path of a projectile where the height is given by a quadratic expression, such as \(h(t) = -t^2 + 7t - 12\), you can find the times at which the object hits the ground (height = 0) by factoring the equation \(0 = -(t-3)(t-4)\). This allows you to quickly determine that the projectile hits the ground at \(t=3\) and \(t=4\) seconds. Understanding how to factor these equations helps in fields ranging from engineering and physics to economics.