To solve for $x$, we are given that the two angles are complementary. This means their measures add up to $90^\circ$.

Let the two angles be:

• First angle: $x^2 - 6x$
• Second angle: $3x + 80$

The sum of these two angles is $90^\circ$, so we can set up the equation:

$(x^2 - 6x) + (3x + 80) = 90$

Now, simplify and solve for $x$:

1. Combine like terms: $x^2 - 6x + 3x + 80 = 90$ $x^2 - 3x + 80 = 90$

2. Subtract 90 from both sides to set the equation equal to zero: $x^2 - 3x + 80 - 90 = 0$ $x^2 - 3x - 10 = 0$

3. Now, solve the quadratic equation $x^2 - 3x - 10 = 0$ by factoring. We look for two numbers that multiply to $-10$ and add up to $-3$. These numbers are $-5$ and $2$.

So, the factored form is: $(x - 5)(x + 2) = 0$

4. Set each factor equal to zero: $x - 5 = 0 \quad \text{or} \quad x + 2 = 0$

5. Solve for $x$: $x = 5 \quad \text{or} \quad x = -2$

Thus, the possible values of $x$ are $x = 5$ and $x = -2$.

To verify:

• If $x = 5$:

  • First angle: $5^2 - 6(5) = 25 - 30 = -5^\circ$
  • Second angle: $3(5) + 80 = 15 + 80 = 95^\circ$

  This gives a contradiction since angles cannot be negative, so $x = 5$ is not a valid solution.

• If $x = -2$:

  • First angle: $(-2)^2 - 6(-2) = 4 + 12 = 16^\circ$
  • Second angle: $3(-2) + 80 = -6 + 80 = 74^\circ$

  These two angles add up to $16^\circ + 74^\circ = 90^\circ$, which satisfies the condition of complementary angles.

So, the only valid solution is $x = -2$.

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Would you like further clarification or have any questions? Here are 5 related questions you can explore:

1. What is the difference between complementary and supplementary angles?
2. How do you solve quadratic equations by completing the square?
3. What is the definition of an angle in geometry?
4. How can you check the solutions to a system of equations?
5. How do you apply factoring in solving polynomials?

Tip: Always check the physical meaning of the problem (e.g., angles cannot be negative) to ensure the solution is valid in context.