Answer: The required values are \(\dfrac{169}{4}~~\textup{and}~~\dfrac{157}{4}.\)
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
\(x^2+13x=-3\\\\\Rightarrow x^2+2\times x\times \dfrac{13}{2}+\left(\dfrac{13}{2}\right)^2=\left(\dfrac{13}{2}\right)^2-3\\\\\\\Rightarrow \left(x+\dfrac{13}{2}\right)^2=\dfrac{169}{4}-3\\\\\\\Rightarrow \left(x+\dfrac{13}{2}\right)^2=\dfrac{157}{4}.\)
So, the complete statement is
\(x^2+13x+\dfrac{169}{4}=\dfrac{157}{4}.\)
Thus, the required values are \(\dfrac{169}{4}~~\textup{and}~~\dfrac{157}{4}.\)
