What are the solutions of [tex]x^2-6 x-72=0[/tex]?
A. [tex]x=12,6[/tex]
B. [tex]x=12,-6[/tex]
C. [tex]x=13,-6[/tex]
D. [tex]x=-12,-6[/tex]
Answer (2)
Factor the quadratic equation x 2 − 6 x − 72 = 0 to get ( x − 12 ) ( x + 6 ) = 0 .
Set each factor to zero: x − 12 = 0 or x + 6 = 0 .
Solve for x : x = 12 or x = − 6 .
The solutions are x = 12 , − 6 .
Explanation
Understanding the Problem We are asked to find the solutions to the quadratic equation x 2 − 6 x − 72 = 0 . This means we need to find the values of x that make the equation true. We can solve this by factoring the quadratic expression.
Factoring the Quadratic To factor the quadratic expression x 2 − 6 x − 72 , we need to find two numbers that multiply to -72 and add up to -6. Let's think of factors of 72: 1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9. Since the product is negative, one factor must be positive and the other negative. Since the sum is -6, the larger factor must be negative. The pair 6 and 12 works, so we use -12 and 6 since − 12 × 6 = − 72 and − 12 + 6 = − 6 .
Rewriting the Equation Now we can rewrite the quadratic equation in factored form: ( x − 12 ) ( x + 6 ) = 0 .
Setting Factors to Zero To find the solutions, we set each factor equal to zero: x − 12 = 0 or x + 6 = 0 .
Finding the Solutions Solving for x in each equation, we get: x = 12 or x = − 6 . Therefore, the solutions to the quadratic equation are x = 12 and x = − 6 .
Examples
Quadratic equations are used in many real-world applications, such as calculating the trajectory of a ball, designing bridges, and determining the optimal dimensions for a garden. For example, if you want to build a rectangular garden with an area of 72 square feet and you know that the length must be 6 feet longer than the width, you can use a quadratic equation to find the dimensions of the garden.
The solutions to the quadratic equation x 2 − 6 x − 72 = 0 are x = 12 and x = − 6 . This is obtained by factoring the quadratic into ( x − 12 ) ( x + 6 ) = 0 and solving each factor. Therefore, the correct answer is B. x = 12 , − 6 .
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