Solve 74x - 2x^2 = 690.

Step-by-step solution

1. 1

   Rearrange into standard form. Move everything to one side so the quadratic machinery applies:

   $-2x^2+74x-690=0$

2. 2

   Normalise the coefficients. Multiply by $-1$ to make the leading coefficient positive, then divide by $2$:

   $2x^2-74x+690=0\qquad\Longrightarrow\qquad x^2-37x+345=0$

   Both operations are legal on an equation and make the numbers as small as they can be.

3. 3

   Look for integer factors, and find none. Factoring would need two numbers with product $345$ and sum $37$. Since $345=3\times5\times23$, the factor pairs are $(1,345),(3,115),(5,69),(15,23)$, with sums $346,118,74,38$ — none is $37$. (Note how close $15+23=38$ comes: the problem is only just unsolvable.)

4. 4

   Compute the discriminant.

   $\Delta=(-37)^2-4(1)(345)=1369-1380=-11$

   Negative, so there are no real solutions. The parabola $x^2-37x+345$ sits entirely above the axis.

5. 5

   Explain the near miss. The left side $74x-2x^2$ is a downward parabola peaking at $x=\tfrac{74}{4}=18.5$, where its value is $74(18.5)-2(18.5)^2=1369-684.5=684.5$. Since the maximum possible value is $684.5$ and the target is $690$, the equation asks for something $5.5$ beyond reach. That shortfall is exactly what the $-11$ discriminant records (the two differ by the factor $2$ from the earlier division).

6. 6

   Give the complex roots and verify. Over $\mathbb{C}$:

   $x=\frac{37\pm i\sqrt{11}}{2}$

   Substituting either into $74x-2x^2$ returns $690$ to within $10^{-9}$ ✓, and a scan of $200{,}001$ real values of $x$ from $-1000$ to $1000$ finds no real solution ✓.