Solve the quadratic equation numerically (using tables of [tex]$x$[/tex]- and [tex]$y$[/tex]-values).
[tex]$x^2+7 x+12=0$[/tex]
a. [tex]$x=-1$[/tex] or [tex]$x=-1$[/tex]
c. [tex]$x=-3$[/tex] or [tex]$x=-3$[/tex]
b. [tex]$x=-4$[/tex] or [tex]$x=-3$[/tex]
d. [tex]$x=2$[/tex] or [tex]$x=-1$[/tex]
Please select the best answer from the choices provided
A
B
C
D
Answer (2)
The quadratic equation x 2 + 7 x + 12 = 0 can be factored into ( x + 3 ) ( x + 4 ) = 0 , leading to the solutions x = − 3 and x = − 4 . Therefore, the correct option selected from the choices is B: x = − 4 or x = − 3 .
;
Factor the quadratic equation x 2 + 7 x + 12 = 0 into ( x + 3 ) ( x + 4 ) = 0 .
Set each factor to zero: x + 3 = 0 and x + 4 = 0 .
Solve for x to find the roots: x = − 3 and x = − 4 .
The solution to the quadratic equation is x = − 4 or x = − 3 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 7 x + 12 = 0 . Our goal is to find the values of x that satisfy this equation. These values are also known as the roots or solutions of the quadratic equation.
Factoring the Quadratic Equation To solve the quadratic equation, we can try to factor it. We are looking for two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4. Therefore, we can rewrite the equation as ( x + 3 ) ( x + 4 ) = 0 .
Finding the Roots Now, we set each factor equal to zero and solve for x . If x + 3 = 0 , then x = − 3 . If x + 4 = 0 , then x = − 4 . Thus, the roots of the quadratic equation are x = − 3 and x = − 4 .
Selecting the Correct Answer Comparing our solution with the given options, we find that option b, x = − 4 or x = − 3 , matches our solution.
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, determining the dimensions of a rectangular area with a given perimeter and area, or modeling the growth of a population. For instance, if you want to build a rectangular garden with an area of 12 square meters and a perimeter that requires 14 meters of fencing, you can use a quadratic equation to find the length and width of the garden. Let l be the length and w be the width. Then l × w = 12 and 2 ( l + w ) = 14 , which simplifies to l + w = 7 . From this, w = 7 − l . Substituting into the area equation, we get l ( 7 − l ) = 12 , which gives 7 l − l 2 = 12 , or l 2 − 7 l + 12 = 0 . Solving this quadratic equation gives l = 3 or l = 4 . If l = 3 , then w = 4 , and if l = 4 , then w = 3 . So the dimensions of the garden are 3 meters and 4 meters.
