We are tasked with solving for two numbers whose sum of squares is 356 and difference is 6.

Let the two numbers be $x$ and $y$.

1. From the problem, we have the equations: x^2 + y^2 = 356 \tag{1} x - y = 6 \tag{2}

Step 1: Express $x$ in terms of $y$ using Equation (2)

From Equation (2): x = y + 6 \tag{3}

Step 2: Substitute $x$ from Equation (3) into Equation (1)

Substitute $x = y + 6$ into $x^2 + y^2 = 356$: $(y + 6)^2 + y^2 = 356$

Expand $(y + 6)^2$: $y^2 + 12y + 36 + y^2 = 356$

Combine like terms: $2y^2 + 12y + 36 = 356$

Simplify: $2y^2 + 12y - 320 = 0$

Divide through by 2 to simplify further: y^2 + 6y - 160 = 0 \tag{4}

Step 3: Solve the quadratic equation (4)

The quadratic equation $y^2 + 6y - 160 = 0$ can be solved using the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 6$, and $c = -160$.

Substitute into the formula: $y = \frac{-6 \pm \sqrt{6^2 - 4(1)(-160)}}{2(1)}$ $y = \frac{-6 \pm \sqrt{36 + 640}}{2}$ $y = \frac{-6 \pm \sqrt{676}}{2}$ $y = \frac{-6 \pm 26}{2}$

Step 4: Calculate the two possible values of $y$

$y = \frac{-6 + 26}{2} = \frac{20}{2} = 10$ $y = \frac{-6 - 26}{2} = \frac{-32}{2} = -16$

Step 5: Find $x$ for each value of $y$

Using $x = y + 6$:

1. If $y = 10$, then $x = 10 + 6 = 16$.
2. If $y = -16$, then $x = -16 + 6 = -10$.

Final Answer:

The two numbers are:

1. $x = 16$, $y = 10$, or
2. $x = -10$, $y = -16$.

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Do you want a detailed explanation of how the quadratic formula was applied? Here are some follow-up questions for practice:

1. What happens if the difference between the two numbers changes to 10?
2. Can you derive a different pair of equations to solve the problem?
3. What is the geometric interpretation of the sum of squares?
4. How does the discriminant in the quadratic formula affect the solution?
5. Can you solve the problem graphically?

Tip: Always verify solutions by substituting them back into the original equations!