Solve the equation (x + 8)(x + 3) = 9.

Step-by-step solution

1. 1

   Do not use the zero-product rule yet. $(x+8)(x+3)=9$ does not mean $x+8=9$ or $x+3=9$. The zero-product property works only when the product equals zero, so the equation must first be brought to standard form.

2. 2

   Expand and collect.

   $(x+8)(x+3)=x^2+3x+8x+24=x^2+11x+24$

   $x^2+11x+24=9\quad\Longrightarrow\quad x^2+11x+15=0$

3. 3

   Test whether it factors over the integers. We need two integers with product $15$ and sum $11$. The only factor pairs of $15$ are $1\cdot 15$ (sum $16$) and $3\cdot 5$ (sum $8$); neither sums to $11$. In particular the tempting pair $6$ and $5$ fails, since $6\cdot 5=30\neq 15$. So this quadratic has no integer factorisation.

4. 4

   Confirm with the discriminant, then apply the quadratic formula.

   $\Delta=11^2-4(1)(15)=121-60=61$

   $61$ is prime, hence not a perfect square, which proves the roots are irrational:

   $x=\frac{-11\pm\sqrt{61}}{2}$

5. 5

   Evaluate and check numerically. $\sqrt{61}\approx 7.8102$, so

   $x\approx-1.5949\qquad\text{or}\qquad x\approx-9.4051$

   Substituting $x\approx-1.5949$: $(6.4051)(1.4051)\approx 9.000$ $\checkmark$. Vieta gives a second check: the roots sum to $-11$ and multiply to $15$, matching the coefficients.