Solve (8x+9)(5-x)=31x+94: Distributive Property Application

To solve the equation $(8x+9)(5-x) = 31x + 94$, we will use the distributive property. The steps are as follows:

• Step 1: Apply the distributive property to the left-hand side.
  We do this by multiplying each term in the first binomial by each term in the second one:
• $(8x + 9)(5 - x) = 8x \cdot 5 + 8x \cdot (-x) + 9 \cdot 5 + 9 \cdot (-x)$.
• Simplify the terms:
  $8x \cdot 5 = 40x$,
  $8x \cdot (-x) = -8x^2$,
  $9 \cdot 5 = 45$,
  $9 \cdot (-x) = -9x$.
• Combine all like terms:
  The left-hand side becomes $-8x^2 + 40x + 45 - 9x = -8x^2 + 31x + 45$.
• Step 2: Compare both sides of the equation:
  We have $-8x^2 + 31x + 45$ on the left-hand side and $31x + 94$ on the right.
• Step 3: Set the equation:
  $-8x^2 + 31x + 45 = 31x + 94$.
• Step 4: Subtract $31x$ from both sides:
  $-8x^2 + 45 = 94$.
• Step 5: Now subtract 45 from both sides:
  $-8x^2 = 49$.
• Step 6: Divide by $-8$:
  $x^2 = -\frac{49}{8}$.
• Step 7: Evaluate the Result:
  The result $x^2 = -\frac{49}{8}$ implies taking the square root of a negative number, which results in an imaginary number, thus indicating no real solution exists.

Therefore, the solution to the equation is: There is no solution to the equation.