Solve by factoring or finding square roots. [tex]x^2-6 x-7=0[/tex]
Answer (2)
Factor the quadratic expression: x 2 − 6 x − 7 = ( x − 7 ) ( x + 1 ) .
Set each factor to zero: x − 7 = 0 or x + 1 = 0 .
Solve for x : x = 7 or x = − 1 .
The solutions are 7 , − 1 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 6 x − 7 = 0 . Our goal is to find the values of x that satisfy this equation by factoring the quadratic expression.
Factoring the Quadratic We need to factor the quadratic expression x 2 − 6 x − 7 . We are looking for two numbers that multiply to -7 and add up to -6. These numbers are -7 and 1.
Rewriting the Equation So, we can rewrite the quadratic equation as ( x − 7 ) ( x + 1 ) = 0 .
Solving for x Now, we set each factor equal to zero and solve for x :
x − 7 = 0 or x + 1 = 0
Solving these equations gives us:
x = 7 or x = − 1
Final Answer Therefore, the solutions to the quadratic equation x 2 − 6 x − 7 = 0 are x = 7 and x = − 1 .
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For example, engineers use them to design bridges and calculate the trajectory of projectiles. Imagine you're launching a rocket; quadratic equations can help you determine its path, ensuring it reaches its target accurately. They also appear in economics, helping to model cost and revenue curves to optimize business decisions. Understanding quadratic equations opens doors to solving many practical problems in science, engineering, and economics.
To solve the equation x 2 − 6 x − 7 = 0 , we factor it as ( x − 7 ) ( x + 1 ) = 0 . This gives us the solutions x = 7 and x = − 1 . Thus, the final solutions are 7 and − 1 .
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