Question

solve for x.
$4x^2 + 7x + 3 = 0$
$x = ?, -\frac{\quad}{\quad}$
enter the smallest solution first.

Explanation:

Step1: Factor the quadratic equation

We factor $4x^2+7x+3$ by finding two numbers that multiply to $4\times3=12$ and add to $7$ (these numbers are 3 and 4). Split the middle term and factor:

$$\begin{align*} 4x^2+4x+3x+3&=0\\ 4x(x+1)+3(x+1)&=0\\ (4x+3)(x+1)&=0 \end{align*}$$

Step2: Set each factor to zero

Solve for $x$ from each factor:

Substep2a: Solve $4x+3=0$

$4x = -3 \implies x = -\frac{3}{4}$

Substep2b: Solve $x+1=0$

$x = -1$

Step3: Order the solutions

Compare $-1$ and $-\frac{3}{4}$: $-1 < -\frac{3}{4}$

Answer:

$x = -1, -\frac{3}{4}$