[tex]x^2-7 x+12=0[/tex]
Answer (2)
Factor the quadratic expression: x 2 − 7 x + 12 = ( x − 3 ) ( x − 4 ) .
Set each factor equal to zero: x − 3 = 0 or x − 4 = 0 .
Solve for x : x = 3 or x = 4 .
The solutions are x = 3 , 4 .
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. We will solve this by factoring the quadratic expression.
Factoring the Quadratic To factor the quadratic expression x 2 − 7 x + 12 , we need to find two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4, since ( − 3 ) \t \t × ( − 4 ) = 12 and ( − 3 ) + ( − 4 ) = − 7 .
Rewriting the Equation Now we can rewrite the quadratic equation in factored form: ( x − 3 ) ( x − 4 ) = 0 .
Setting Factors to Zero To find the solutions for x , we set each factor equal to zero: x − 3 = 0 or x − 4 = 0 .
Solving for x Solving for x in each equation, we get: x = 3 or x = 4 .
Final Answer Therefore, the solutions to the quadratic equation x 2 − 7 x + 12 = 0 are x = 3 and x = 4 .
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, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 12 square meters and you have 7 meters of fencing, you can use the equation x 2 − 7 x + 12 = 0 to find the possible lengths of the sides of the garden. The solutions x = 3 and x = 4 represent the possible lengths of the sides.
The solutions to the quadratic equation x 2 − 7 x + 12 = 0 are found by factoring it into ( x − 3 ) ( x − 4 ) = 0 . Setting each factor to zero results in x = 3 and x = 4 . Therefore, the values of x that satisfy the equation are 3 and 4.
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