Solve for $x$:
$x^2+17 x+72=0$
Box1 (Larger Solution): $\square$
Box2 (Smaller Solution): $\square$
Answer (1)
Factor the quadratic equation x 2 + 17 x + 72 = 0 into ( x + 8 ) ( x + 9 ) = 0 .
Set each factor equal to zero: x + 8 = 0 or x + 9 = 0 .
Solve for x to find the solutions: x = − 8 and x = − 9 .
Identify the larger and smaller solutions: − 8 and − 9 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 17 x + 72 = 0 . Our goal is to find the values of x that satisfy this equation. There are a couple of ways we can approach this: factoring or using the quadratic formula. Let's try factoring first, as it's often quicker if it works!
Factoring the Quadratic To factor the quadratic, we need to find two numbers that multiply to 72 (the constant term) and add up to 17 (the coefficient of the x term). Let's think about the factors of 72: 1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9. Aha! 8 and 9 add up to 17. So, we can rewrite the quadratic equation as: ( x + 8 ) ( x + 9 ) = 0
Finding the Solutions Now, for the product of two factors to be zero, at least one of them must be zero. This gives us two possible equations:
x + 8 = 0 or x + 9 = 0
Solving these equations for x , we get:
x = − 8 or x = − 9
Identifying Larger and Smaller Solutions So, the solutions to the quadratic equation are x = − 8 and x = − 9 . Now we need to determine which one is larger and which one is smaller. On the number line, -8 is to the right of -9, so -8 is larger than -9.
Therefore, the larger solution is -8 and the smaller solution is -9.
Final Answer Finally, we can fill in the boxes:
Box1 (Larger Solution): -8 Box2 (Smaller Solution): -9
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, imagine you're designing a rectangular garden. You know you want the area to be 72 square feet, and you want the length to be 17 feet longer than the width. By setting up a quadratic equation, you can determine the exact dimensions (length and width) of the garden. Similarly, quadratic equations are used in physics to model projectile motion, in finance to calculate compound interest, and in engineering to design structures.
