Question

Let ​ x^2+13x=−3 ​ .

What values make an equivalent number sentence after completing the square?

Enter your answers in the boxes.

x2+13x+___=____

Answer 1

Answer: The required values are
`(169)/(4)~~\textup{and}~~(157)/(4).`

Step-by-step explanation: The given quadratic equation is

`x^2+13x=-3~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

We are to find the values that make an equivalent number sentence after completing the square.

COMPLETING THE SQUARE : Its a method of making one side of a quadratic equation a perfect square, so that the equation can be solved easily by taking the square root on both the sides of the equation.

From equation (i), we have

`x^2+13x=-3\\\\\Rightarrow x^2+2* x* (13)/(2)+\left((13)/(2)\right)^2=\left((13)/(2)\right)^2-3\\\\\\\Rightarrow \left(x+(13)/(2)\right)^2=(169)/(4)-3\\\\\\\Rightarrow \left(x+(13)/(2)\right)^2=(157)/(4).`

So, the complete statement is

`x^2+13x+(169)/(4)=(157)/(4).`

Thus, the required values are
`(169)/(4)~~\textup{and}~~(157)/(4).`