Solve the quadratic by factoring. Show your work.
[tex]x^2+8=-6 x[/tex]
Answer (2)
Rewrite the equation in standard form: x 2 + 6 x + 8 = 0 .
Find two numbers that multiply to 8 and add up to 6 (2 and 4).
Factor the quadratic: ( x + 2 ) ( x + 4 ) = 0 .
Solve for x : x = − 2 or x = − 4 . The solutions are − 2 , − 4 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 8 = − 6 x and we need to solve it by factoring.
Rewrite in Standard Form First, we need to rewrite the equation in the standard form a x 2 + b x + c = 0 . Adding 6 x to both sides of the equation, we get: x 2 + 6 x + 8 = 0
Find the Factors Now, we need to find two numbers that multiply to 8 (the constant term) and add up to 6 (the coefficient of the x term). Let's list the factors of 8:
1 and 8
2 and 4 The pair 2 and 4 add up to 6, so these are the numbers we're looking for.
Factor the Quadratic We can now factor the quadratic equation using the two numbers we found: ( x + 2 ) ( x + 4 ) = 0
Set Each Factor to Zero To find the solutions for x , we set each factor equal to zero: x + 2 = 0 or x + 4 = 0
Solve for x Solving these linear equations, we get: x = − 2 or x = − 4 So the solutions to the quadratic equation are x = − 2 and x = − 4 .
Examples
Factoring quadratic equations is a fundamental skill in algebra and has many real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as the path of a ball thrown through the air or the flight of a rocket. Architects use quadratic equations to design arches and other curved structures. Financial analysts use quadratic equations to model investment growth and predict future profits. Understanding how to solve quadratic equations by factoring provides a powerful tool for solving problems in a variety of fields.
To solve the quadratic equation x 2 + 8 = − 6 x , we first rearranged it to standard form x 2 + 6 x + 8 = 0 and then factored it as ( x + 2 ) ( x + 4 ) = 0 . This gives the solutions x = − 2 and x = − 4 .
;
