2. [tex]x^2=x+6[/tex]
Answer (1)
Rewrite the equation in standard quadratic form: x 2 − x − 6 = 0 .
Factor the quadratic expression: ( x − 3 ) ( x + 2 ) = 0 .
Set each factor equal to zero: x − 3 = 0 or x + 2 = 0 .
Solve for x : x = 3 or x = − 2 . The solutions are 3 , − 2 .
Explanation
Problem Analysis We are given the quadratic equation x 2 = x + 6 . Our goal is to find the values of x that satisfy this equation.
Rewrite in Standard Form First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . Subtracting x and 6 from both sides of the equation, we get: x 2 − x − 6 = 0
Factor the Quadratic Now, we need to factor the quadratic expression x 2 − x − 6 . We are looking for two numbers that multiply to − 6 and add to − 1 . These numbers are − 3 and 2 . Therefore, we can factor the expression as: ( x − 3 ) ( x + 2 ) = 0
Set Factors to Zero Next, we set each factor equal to zero and solve for x :
x − 3 = 0 or x + 2 = 0
Solve for x Solving for x in each case, we get: x = 3 or x = − 2 Thus, the solutions to the quadratic equation are x = 3 and x = − 2 .
Final Answer Therefore, the solutions to the equation x 2 = x + 6 are x = 3 and x = − 2 .
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a ball, determining the dimensions of a garden, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 6 square meters and the length must be 1 meter longer than the width, you can use a quadratic equation to find the dimensions of the garden. In this case, if x is the width, then the length is x + 1 , and the area is x ( x + 1 ) = 6 , which simplifies to x 2 + x − 6 = 0 . Solving this equation gives you the possible values for the width of the garden.
