Question

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Which of the following tables shows the correct steps to transform x2 + 10x + 24 = 0 into the form (x − p)2 = q?
[p and q are integers]


Step 1 x2 + 10x + 24 − 1 = 0 − 1
Step 2 x2 + 10x + 23 = −1
Step 3 (x + 5)2 = −1

Step 1 x2 + 10x + 24 − 2 = 0 − 2
Step 2 x2 + 10x + 22 = −2
Step 3 (x + 5)2 = −2

Step 1 x2 + 10x + 24 + 2 = 0 + 2
Step 2 x2 + 10x + 26 = 2
Step 3 (x + 5)2 = 2

Step 1 x2 + 10x + 24 + 1 = 0 + 1
Step 2 x2 + 10x + 25 = 1
Step 3 (x + 5)2 = 1

Answer 1

The process of turning a quadratic equation into a perfect square is called completing the square

x2 + 10x + 24 = 0

Take the x term (10x). The coefficient is 10. Take this number, divide it by 2, square it. 10/2 = 5, 5^2 = 25

So x2 + 10x + 25 would be a perfect square trinomial. x2 + 10x + 25 = (x + 5)^2

However, we have x^2 + 10x + 24 instead of + 25. So we add 1 to both sides to get

x^2 + 10x + 24 + 1 = 0 + 1

And rewrite the expression from there, converting the left side to (x+5)^2 and simplifying the right side to 1