Solve the quadratic equation. [tex]x^2-7 x=30[/tex]
Answer (2)
To solve the equation x 2 − 7 x = 30 , rewrite it as x 2 − 7 x − 30 = 0 and factor it to obtain ( x − 10 ) ( x + 3 ) = 0 . The solutions are x = 10 and x = − 3 .
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Rewrite the equation in standard form: x 2 − 7 x − 30 = 0 .
Factor the quadratic expression: ( x − 10 ) ( x + 3 ) = 0 .
Set each factor to zero and solve for x : x − 10 = 0 or x + 3 = 0 .
The solutions are: 10 and − 3 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 7 x = 30 . Our goal is to find the values of x that satisfy this equation.
Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . To do this, we subtract 30 from both sides of the equation: x 2 − 7 x − 30 = 0
Factoring the Quadratic Now we need to solve the quadratic equation x 2 − 7 x − 30 = 0 . We can solve this by factoring. We are looking for two numbers that multiply to -30 and add up to -7. These numbers are -10 and 3, since ( − 10 ) × 3 = − 30 and − 10 + 3 = − 7 . Therefore, we can factor the quadratic equation as follows: ( x − 10 ) ( x + 3 ) = 0
Finding the Solutions To find the solutions for x , we set each factor equal to zero: x − 10 = 0 or x + 3 = 0
The Solutions Solving for x in each case, we get: x = 10 or x = − 3
Final Answer Therefore, the solutions to the quadratic equation x 2 − 7 x = 30 are x = 10 and x = − 3 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, or modeling the growth of a population. For instance, if you want to build a rectangular garden with an area of 30 square meters and you know that the length must be 7 meters longer than the width, you can use a quadratic equation to find the dimensions of the garden. Let w be the width and l be the length. We have l = w + 7 and l × w = 30 . Substituting the first equation into the second, we get ( w + 7 ) w = 30 , which simplifies to w 2 + 7 w − 30 = 0 . Solving this quadratic equation gives us the width of the garden.
