To complete the equation \( 4x + 10 = x^2 \) so that it has infinite solutions, we need to set it up in a way that the two sides of the equation are identical for all values of \( x \).
Let's manipulate the equation:
-
Start with the equation: \[ 4x + 10 = x^2 \]
-
Rearranging the equation gives: \[ x^2 - 4x - 10 = 0 \]
For the equation to have infinite solutions, it needs to be an identity, meaning the two sides must be equal for all \( x \). One way this can happen is if we have both sides of the equation equal to zero.
To ensure that the quadratic \( x^2 - 4x - 10 \) is identically zero for all \( x \), both coefficients must be zero, leading to:
- \( 1 = 0 \) (coefficient of \( x^2 \)),
- \( -4 = 0 \) (coefficient of \( x \)),
- \( -10 = 0 \) (constant term).
However, since these conditions cannot realistically hold (a non-zero polynomial cannot be identically zero), we can adjust the original equation in another way to create dependent variables on either side.
Instead, let's rewrite the right side to directly reflect the left side:
If you want an identity: \[ 4x + 10 = 4x + 10 \]
Then, the complete equation that has infinite solutions can be written as: \[ 4x + 10 = 4x + 10 \]
This ensures that for any value of \( x \), both sides remain equal, thus providing infinite solutions.