Solve 4.08x^2 - 652.8x + 10236 = 0.

Step-by-step solution

1. 1

   Identify the coefficients and pick the method. Here $a=4.08$, $b=-652.8$, $c=10236$. The decimals rule out factoring by inspection, so use

   $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

2. 2

   Compute the discriminant one product at a time. Each piece is an exact decimal, so no rounding is needed yet:

   $b^{2}=652.8^{2}=426147.84,\qquad 4ac=4(4.08)(10236)=16.32\times 10236=167051.52$

   $b^{2}-4ac=426147.84-167051.52=259096.32$

   It is positive, so there are two distinct real roots.

3. 3

   Take the square root.

   $\sqrt{259096.32}=509.0150489\ldots$

   Close to $509$, but not equal to it — rounding here to a flat $509$ shifts both roots in the third decimal place, so keep the extra digits.

4. 4

   Substitute into the formula. With $2a=8.16$ and $-b=652.8$:

   $x=\frac{652.8\pm 509.0150489}{8.16}$

   $x_{1}=\frac{1161.8150489}{8.16}\approx 142.37930,\qquad x_{2}=\frac{143.7849511}{8.16}\approx 17.62070$

5. 5

   Recover the exact form by clearing decimals. Multiplying the original equation by $25$ and dividing by $6$ gives the integer equation $17x^{2}-2720x+42650=0$, whose discriminant is $4498200=2^{3}3^{3}5^{2}7^{2}\cdot 17$, so $\sqrt{4498200}=210\sqrt{102}$ and

   $x=\frac{1360\pm 105\sqrt{102}}{17}$

6. 6

   Check with Vieta's formulas. The sum of the roots must be $-\dfrac{b}{a}=\dfrac{652.8}{4.08}=160$, and indeed $142.37930+17.62070=160.00000$. The product must be $\dfrac{c}{a}=\dfrac{10236}{4.08}=\dfrac{42650}{17}=2508.8235$, and $142.37930\times 17.62070=2508.8235\ \checkmark$