Question

What number needs to be added to both sides of the equation in order to complete the square? 1) +9, +9 2) -9, -9 3) +9, -9 4) -9, +9

Answer 1

The option is 4) -9, +9. To complete the square, a number must be added to both sides of the equation, and this number is the additive inverse of the coefficient of the linear term.

Step-by-step explanation:

Completing the square is a process used to convert a quadratic expression into a perfect square trinomial. In the quadratic equation
`\(ax^2 + bx + c = 0\)`, the key step is to add
`\(\left((b)/(2)\right)^2\)` to both sides. This term ensures that the quadratic expression becomes a perfect square trinomial, facilitating the use of the square root property for solving equations.

For the equation
`\(ax^2 + bx + c = 0\),` the term
`\(\left((b)/(2)\right)^2\)` is
`\((b^2)/(4)\)`, and to add this term to both sides, the additive inverse
`\(-(b^2)/(4)\)` is subtracted from both sides. The result is a perfect square trinomial on one side, and the equation can be factored into
`\((x + (b)/(2))^2 = (b^2 - 4ac)/(4a)\)`. The square root property can then be applied to find the solutions.

In the provided options, -9 is the additive inverse of 9, which is the coefficient of the linear term. Therefore, adding -9 to both sides completes the square for the given quadratic equation.

In summary, to complete the square for a quadratic equation, the additive inverse of the coefficient of the linear term must be added to both sides. In this case, the correct option is 4) -9, +9.