Solve the equation 40x + 30x + 30x + 2x^2 = 450.

Step-by-step solution

1. 1

   Combine the linear terms. All three share the same power of $x$:

   $40x + 30x + 30x = 100x \quad\Longrightarrow\quad 2x^2 + 100x = 450$

2. 2

   Move everything to one side and divide by the common factor.

   $2x^2 + 100x - 450 = 0 \quad\Longrightarrow\quad x^2 + 50x - 225 = 0$

   Dividing by $2$ makes the quadratic monic, which simplifies both the factor search and the formula.

3. 3

   Test whether it factors over the integers. Factoring would need two integers with product $-225$ and sum $50$. The divisor pairs of $225$ are $(1,225), (3,75), (5,45), (9,25), (15,15)$, giving possible sums $\pm224, \pm72, \pm40, \pm16, 0$ — none is $50$. So no integer factorisation exists and the quadratic formula is required.

4. 4

   Apply the quadratic formula. With $a = 1$, $b = 50$, $c = -225$:

   $b^2 - 4ac = 2500 + 900 = 3400, \qquad \sqrt{3400} = 10\sqrt{34} \approx 58.309519$

   $x = \frac{-50 \pm 10\sqrt{34}}{2} = -25 \pm 5\sqrt{34}$

5. 5

   Evaluate both roots.

   $x_1 = -25 + 5\sqrt{34} \approx 4.154759, \qquad x_2 = -25 - 5\sqrt{34} \approx -54.154759$

6. 6

   Verify, and reject a common wrong answer. For $x_1$: $2(4.154759)^2 + 100(4.154759) = 34.52406 + 415.47590 = 449.99994$. For $x_2$: $5865.47584 - 5415.47590 = 449.99994$ (the small shortfall is only the rounding of $\sqrt{34}$). Vieta also checks — sum $= -50$, product $= -225$. The frequently seen answer $x = 5$ or $x = -55$ is wrong: both give $2x^2 + 100x = 550$, not $450$, because $55 \times (-5) = -275$ rather than $-225$.