If using the method of completing the square to solve the quadratic equation x^2 – 14x – 40 = 0, which number would have to be added to “complete the square”
The Correct Answer and Explanation is:
To solve the quadratic equation x2−14x−40=0x^2 – 14x – 40 = 0x2−14x−40=0 by completing the square, we first need to isolate the terms involving xxx. Start by moving the constant term to the other side of the equation:x2−14x=40x^2 – 14x = 40×2−14x=40
The next step is to “complete the square.” This means turning the left-hand side into a perfect square trinomial. To do that, we take half of the coefficient of xxx, square it, and add that value to both sides of the equation. The coefficient of xxx is -14. Half of -14 is -7, and squaring -7 gives:(−7)2=49(-7)^2 = 49(−7)2=49
So, the number we must add to both sides of the equation to complete the square is 49.
Now we add 49 to both sides:x2−14x+49=40+49x^2 – 14x + 49 = 40 + 49×2−14x+49=40+49(x−7)2=89(x – 7)^2 = 89(x−7)2=89
Next, take the square root of both sides:x−7=±89x – 7 = \pm\sqrt{89}x−7=±89
Solving for xxx, we get:x=7±89x = 7 \pm \sqrt{89}x=7±89
Explanation
Completing the square is a method for solving quadratic equations that allows us to transform a standard quadratic expression into a perfect square trinomial. A quadratic expression has the general form ax2+bx+cax^2 + bx + cax2+bx+c. When the coefficient of x2x^2×2 is 1, the process becomes more straightforward.
In our case, we began with the equation x2−14x−40=0x^2 – 14x – 40 = 0x2−14x−40=0. To complete the square, we first moved the constant -40 to the other side. This isolates the terms involving xxx, making it easier to manipulate the expression.
The heart of completing the square lies in creating a perfect square trinomial from the two terms x2x^2×2 and −14x-14x−14x. To do this, take half the coefficient of xxx, which is -14, giving -7. Squaring -7 results in 49. Adding this number completes the square on the left side, giving us the trinomial x2−14x+49x^2 – 14x + 49×2−14x+49, which factors to (x−7)2(x – 7)^2(x−7)2.
We also add the same value to the right side of the equation to keep it balanced, leading to (x−7)2=89(x – 7)^2 = 89(x−7)2=89. Solving this by taking square roots gives the final solution x=7±89x = 7 \pm \sqrt{89}x=7±89.
This method is particularly useful when factoring is difficult or impossible and is also foundational for deriving the quadratic formula.
