Solve the equation by factoring: $x^2+9 x+20=0$
The problem asks us to solve the quadratic equation \(x^2 + 9x + 20 = 0\) by factoring. A quadratic equation in the form \(ax^2 + bx + c = 0\) can often be factored into \((x + p)(x + q) = 0\), where the product of the constants \(p \times q\) equals the constant term \(c\) (in this case, 20) and the sum of the constants \(p + q\) equals the coefficient of the middle term \(b\) (in this case, 9). Our goal is to find these two integers \(p\) and \(q\).
We need two numbers that multiply to 20 and add to 9. Let's list the factor pairs of 20:
The pair 4 and 5 satisfies both conditions: their product is 20 and their sum is 9. Therefore, we can write the quadratic equation as \((x + 4)(x + 5) = 0\).
Now that the equation is in the form \((x + 4)(x + 5) = 0\), we use the zero product property, which 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:
Solving these equations, we get \(x = -4\) and \(x = -5\). Comparing these results with the given options, we see that they match option C.
Quadratic equations like \(x^2 + 9x + 20 = 0\) are widely used in physics and engineering to model projectile motion. For example, if you throw a ball, its height over time follows a parabolic path defined by a quadratic equation. By solving for the roots, you can determine exactly when the ball hits the ground (the time where height is zero). This mathematical foundation helps engineers predict trajectories and optimize designs for everything from sports equipment to aerospace components.