Solve $x^2+5 x=-6$.
We are given the quadratic equation \(x^2 + 5x = -6\). To solve for \(x\), we need to find the roots of this equation, which are the values of \(x\) that satisfy the equality. We are given four sets of options to choose from: \(\{5, 1\}\), \(\{-5, -1\}\), \(\{3, 2\}\), and \(\{-3, -2\}\).
The first step to solving a quadratic equation is to express it in the standard form \(ax^2 + bx + c = 0\). We can do this by adding 6 to both sides of the given equation:
\[x^2 + 5x + 6 = 0.\]
Now, our equation is in the form where \(a=1\), \(b=5\), and \(c=6\).
To factor the quadratic expression \(x^2 + 5x + 6\), we look for two numbers whose product is \(c=6\) and whose sum is \(b=5\).
Testing pairs of factors for 6:
The correct numbers are 2 and 3. Thus, we can factor the quadratic as:
\[(x+2)(x+3) = 0.\]
According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore:
\(x + 2 = 0 \implies x = -2\)
\(x + 3 = 0 \implies x = -3\)
The solutions to the equation are \(x = -2\) and \(x = -3\). Comparing this result with the provided options, we see that it matches the last option.
Quadratic equations like this are fundamental in fields such as physics and engineering, where they model the trajectories of objects. For example, if you throw a ball, the height of the ball over time can be described by a quadratic equation \(h(t) = at^2 + bt + c\). Setting this expression equal to a specific height (like the ground, where \(h=0\)) requires solving a quadratic equation to determine exactly when the ball will hit that height. Mastering this skill allows you to predict outcomes in dynamic systems and solve various real-world problems.