Carmen is using the quadratic equation (x + 15)(x) = 100 where x represents the width of a picture frame. Which statement about the solutions x = 5 and x = –20 is true?
Can a picture frame be -20?
To determine which statement is true about the solutions x = 5 and x = -20 in the quadratic equation (x + 15)(x) = 100, we should substitute these values into the equation and see if they satisfy it.
Let's start by substituting x = 5 into the equation:
(5 + 15)(5) = 100
(20)(5) = 100
100 = 100
Now, let's substitute x = -20 into the equation:
(-20 + 15)(-20) = 100
(-5)(-20) = 100
100 = 100
Both x = 5 and x = -20 satisfy the equation, so we can conclude that both values are solutions.
Therefore, the true statement about the solutions x = 5 and x = -20 is that both values satisfy the quadratic equation (x + 15)(x) = 100.
To determine which statement about the solutions x = 5 and x = -20 is true for the quadratic equation (x + 15)(x) = 100, we need to solve the equation and then compare the solutions.
Step 1: Expand and simplify the equation:
(x + 15)(x) = 100
x^2 + 15x = 100
Step 2: Rearrange the equation to bring it in the quadratic form:
x^2 + 15x - 100 = 0
Step 3: To solve the quadratic equation, we can use factoring method, completing the square method, or the quadratic formula. In this case, let's use the factoring method.
The equation factors as:
(x + 20)(x - 5) = 0
Step 4: Set each factor equal to zero:
x + 20 = 0 OR x - 5 = 0
Step 5: Solve each equation:
For x + 20 = 0:
x = -20
For x - 5 = 0:
x = 5
Thus, the solutions to the given quadratic equation are x = 5 and x = -20. Now let's analyze the statements.
Statement 1: x = 5 is a solution to the equation.
This statement is true because we found x = 5 to be a solution.
Statement 2: x = -20 is a solution to the equation.
This statement is also true because we found x = -20 to be a solution.
Therefore, both statements are true for the solutions x = 5 and x = -20.