Which of the following equations has exactly the same solutions as (x−7)(x+9)=0 ?(1 point) Responses x2+2x=−63 x squared plus 2 x equals negative 63 x2=−2x+63 x squared equals negative 2 x plus 63 x2−2x−63=0 x squared minus 2 x minus 63 equals 0 x2=2x−63
The equation that has exactly the same solutions as (x−7)(x+9)=0 is x^2−2x−63=0.
To find which equation has exactly the same solutions as (x−7)(x+9)=0, we need to solve this equation and compare the solutions.
Step 1: Solve the equation (x−7)(x+9)=0.
To do this, we set each factor equal to zero and solve for x:
x−7=0 --> x=7
x+9=0 --> x=-9
So, the solutions to the equation (x−7)(x+9)=0 are x=7 and x=-9.
Step 2: Compare the solutions to the given options.
a) x^2 + 2x = −63
To solve this equation, we can move all terms to one side:
x^2 + 2x + 63 = 0
This equation does not have the same solutions as the original equation because the solutions to this equation are not 7 and -9.
b) x^2 = −2x + 63
To solve this equation, we can rearrange terms:
x^2 + 2x - 63 = 0
This equation does not have the same solutions as the original equation because the solutions to this equation are not 7 and -9.
c) x^2 − 2x − 63 = 0
This equation is in the same form as the original equation. Therefore, it has the same solutions as the original equation, which are x=7 and x=-9.
d) x^2 = 2x − 63
To solve this equation, we can rearrange terms:
x^2 - 2x + 63 = 0
This equation does not have the same solutions as the original equation because the solutions to this equation are not 7 and -9.
Therefore, the equation that has exactly the same solutions as (x−7)(x+9)=0 is:
x^2 − 2x − 63 = 0
To find the equation that has the same solutions as (x−7)(x+9)=0, we can use the zero product property. According to the property, if the product of two factors is equal to zero, then at least one of the factors must be equal to zero.
In this case, we have two factors (x−7) and (x+9). So, to find the equation with the same solutions, we need to set each factor equal to zero and solve for x.
Setting (x−7) equal to zero:
x - 7 = 0
x = 7
Setting (x+9) equal to zero:
x + 9 = 0
x = -9
The solutions of the given equation (x−7)(x+9)=0 are x = 7 and x = -9.
Now let's see which equation among the options has the same solutions:
1) x^2 + 2x = -63: The solutions of this equation are not the same as the ones we found earlier.
2) x^2 = -2x + 63: After rearranging, this equation becomes x^2 + 2x - 63 = 0. This equation has the same solutions as the given equation.
3) x^2 - 2x - 63 = 0: This equation has the same solutions as the given equation.
4) x^2 = 2x - 63: After rearranging, this equation becomes x^2 - 2x + 63 = 0. This equation does not have the same solutions as the given equation.
Based on the analysis, the equations that have exactly the same solutions as (x−7)(x+9)=0 are x^2 + 2x - 63 = 0 and x^2 - 2x - 63 = 0.