Solve 4.08x^2 - 652.85x + 10236 = 0 using the quadratic formula. Give both roots to three decimal places.

Step-by-step solution

1. 1

   Identify $a$, $b$ and $c$ with their signs. Comparing with $ax^{2}+bx+c=0$:

   $a=4.08,\qquad b=-652.85,\qquad c=10236.$

   The middle coefficient is negative; carrying that minus sign through $-b$ is what makes both roots come out positive. Since $a>0$ and $c>0$ with $b<0$, the two roots (if real) must both be positive.

2. 2

   Compute $b^{2}$ and $4ac$ separately. Doing the two halves of the discriminant apart from each other makes an arithmetic slip visible:

   $b^{2}=652.85^{2}=426213.1225,$

   $4ac=4\times 4.08\times 10236=16.32\times 10236=167051.52.$

   That second product is the step most often mis-multiplied: $16.32\times10236=163776+3275.52$, and dropping the $0.32\times10236=3275.52$ piece gives a discriminant that is wrong by about $25$.

3. 3

   Form the discriminant and take its square root. Subtracting,

   $\Delta=426213.1225-167051.52=259161.6025,$

   $\sqrt{\Delta}=509.07917\ldots$

   Since $\Delta>0$ there are two distinct real roots — consistent with the sign reasoning in step 1.

4. 4

   Apply the formula with $2a=8.16$. Substituting,

   $x=\frac{652.85\pm 509.07917}{8.16}.$

   The plus branch: $\dfrac{1161.92917}{8.16}=142.39328$. The minus branch: $\dfrac{143.77083}{8.16}=17.61897$.

5. 5

   Check with Vieta's formulas. The sum of the roots must equal $-b/a$ and the product $c/a$:

   $142.39328+17.61897=160.01225=\frac{652.85}{4.08} ✓,$

   $142.39328\times 17.61897=2508.82\ldots=\frac{10236}{4.08} ✓.$

   Both identities hold, which confirms the discriminant arithmetic as well as the division.