Find the solutions of the quadratic equation [tex]x^2+7 x+10=0[/tex]
A) [tex]x=2,5[/tex]
B) [tex]x=-7,-3[/tex]
C) [tex]x=7,3[/tex]
D) [tex]x=-2,-5[/tex]

Factor the quadratic equation: x 2 + 7 x + 10 = ( x + 2 ) ( x + 5 ) .
Set each factor to zero: x + 2 = 0 or x + 5 = 0 .
Solve for x : x = − 2 or x = − 5 .
The solutions are x = − 2 , − 5 ​ .

Explanation

Understanding the Problem We are given the quadratic equation x 2 + 7 x + 10 = 0 and asked to find its solutions. This means we need to find the values of x that satisfy the 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 10 and add up to 7. These numbers are 2 and 5.

Rewriting the Equation We can rewrite the quadratic equation as ( x + 2 ) ( x + 5 ) = 0 .

Solving for x Now, we set each factor equal to zero and solve for x :


x + 2 = 0 or x + 5 = 0
Solving these equations, we get:
x = − 2 or x = − 5

Finding the Solutions Therefore, the solutions to the quadratic equation are x = − 2 and x = − 5 . Comparing these solutions with the given options, we see that option D is the correct one.

Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a ball, determining the dimensions of a rectangular area given its area and perimeter, and modeling various physical phenomena. For example, if you want to build a rectangular garden with an area of 10 square meters and you know that the length must be 3 meters longer than the width, you can set up a quadratic equation to find the dimensions of the garden. Let w be the width and l be the length. Then l = w + 3 and the area is A = l × w = ( w + 3 ) w = 10 . This gives the quadratic equation w 2 + 3 w − 10 = 0 , which can be solved to find the width w and then the length l .

Answered by GinnyAnswer | 2025-07-04