Solve the equation b^2 - 8b + 5 = 0.

Step-by-step solution

1. 1

   Rename the coefficients to avoid a collision. The quadratic formula is usually written with a coefficient called $b$, but here $b$ is the unknown. Use capitals instead:

   $A=1,\qquad B=-8,\qquad C=5,\qquad b=\frac{-B\pm\sqrt{B^2-4AC}}{2A}$

2. 2

   Check for an integer factorisation. Product $5$, sum $-8$: the only pairs are $(-1,-5)$ with sum $-6$ and $(1,5)$ with sum $6$. Neither gives $-8$, so the formula it is.

3. 3

   Compute the discriminant.

   $\Delta=(-8)^2-4(1)(5)=64-20=44$

4. 4

   Substitute and simplify. $44=4\cdot 11$, so $\sqrt{44}=2\sqrt{11}$:

   $b=\frac{8\pm 2\sqrt{11}}{2}=4\pm\sqrt{11}$

5. 5

   Verify with completing the square and numbers. $b^2-8b+5=(b-4)^2-11$, zero when $(b-4)^2=11$ $\checkmark$. Numerically $\sqrt{11}\approx 3.3166$, giving $b\approx 7.3166$ or $b\approx 0.6834$; then $0.6834^2-8(0.6834)+5\approx 0.467-5.467+5\approx 0$ $\checkmark$.