The sum of squares of two consecutive odd indigers is 74.what are  the numbers

Let's solve the problem by setting up an equation.

Let the two consecutive odd integers be \( x \) and \( x + 2 \).

The sum of their squares is given as:
\[
x^2 + (x + 2)^2 = 74
\]

Expand the squared term:
\[
x^2 + (x^2 + 4x + 4) = 74
\]

Combine like terms:
\[
2x^2 + 4x + 4 = 74
\]

Subtract 74 from both sides to set the equation to zero:
\[
2x^2 + 4x - 70 = 0
\]

Now, divide the entire equation by 2 to simplify:
\[
x^2 + 2x - 35 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -35 \).

Calculate the discriminant:
\[
b^2 - 4ac = 2^2 - 4(1)(-35) = 4 + 140 = 144
\]

Now, find the roots:
\[
x = \frac{-2 \pm \sqrt{144}}{2(1)} = \frac{-2 \pm 12}{2}
\]

This gives us two possible solutions:
\[
x = \frac{-2 + 12}{2} = \frac{10}{2} = 5
\]
\[
x = \frac{-2 - 12}{2} = \frac{-14}{2} = -7
\]

Thus, the two sets of consecutive odd integers could be:
1. \( x = 5 \), so the numbers are 5 and 7.
2. \( x = -7 \), so the numbers are -7 and -5.

Both pairs satisfy the condition that their squares sum to 74:
\[
5^2 + 7^2 = 25 + 49 = 74
\]
\[
(-7)^2 + (-5)^2 = 49 + 25 = 74
\]

So, the consecutive odd integers are either \( 5 \) and \( 7 \) or \( -7 \) and \( -5 \).

Would you like more details on any of these steps?

Here are 5 related questions:
1. What is the general formula for the sum of squares of two consecutive integers?
2. How can we factor the quadratic equation \( x^2 + 2x - 35 = 0 \)?
3. What is the significance of the discriminant in solving quadratic equations?
4. How do we solve quadratic equations using the quadratic formula?
5. What are some real-world applications of solving quadratic equations?

**Tip:** When dealing with quadratic equations, always check if they can be factored easily before using the quadratic formula—it can save time!

Learn how to solve the problem of finding two consecutive odd integers whose squares sum up to 74. Understand the application of the quadratic formula and its relevance in solving such problems. Suitable for Grades 7-9.

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