Solve the equation (2x + 3)^2 - 4x^2 = 51 - 9x.

Step-by-step solution

1. 1

   Do not reach for the quadratic formula yet. The equation looks quadratic because of the squared bracket, but the $-4x^2$ sitting outside it is exactly the square of the leading term inside. Expanding is what reveals that, so expand the bracket first:

   $(2x+3)^2=(2x)^2+2\cdot 2x\cdot 3+3^2=4x^2+12x+9$

2. 2

   Substitute and watch the quadratic terms cancel. Put the expansion back into the left-hand side:

   $4x^2+12x+9-4x^2=12x+9$

   The $x^2$ terms are gone, so the whole problem is now linear:

   $12x+9=51-9x$

3. 3

   Collect the $x$ terms on one side and the constants on the other. Add $9x$ to both sides and subtract $9$ from both sides in one move:

   $12x+9x=51-9$

   $21x=42$

4. 4

   Divide by the coefficient of $x$.

   $x=\frac{42}{21}=2$

5. 5

   Check the answer in the original equation, not the simplified one. A cancellation step is exactly where a sign slip hides, so substitute $x=2$ back into the line you started from:

   $\text{LHS}=(2\cdot 2+3)^2-4\cdot 2^2=7^2-16=49-16=33$

   $\text{RHS}=51-9\cdot 2=51-18=33$

   Both sides equal $33$, so $x=2$ is confirmed.