Q 284 - Intermediate Algebra

Question

In the following exercises, solve:

$4b^2+ 7b = -3$

Step-by-Step Solution

Answer

The solutions are $b=\frac{-3}{4},-1.$

Step 1. Given Information

The given quadratic equation is $4b^2+ 7b = -3.$

We have to solve the quadratic equation by factoring.

Step 2. Factor the quadratic expression

First, write the given equation in standard form.

The standard form of the quadratic equation is

$a{x}^2+bx+c=0$

Therefore,

$$\begin{gathered} 4b^2+ 7b = -3 \\ 4b^2+7b+3=0 \end{gathered}$$

Now, let's factor the quadratic expression 

$$\begin{gathered} 4b^2+7b+3=0 \\ 4b^2+4b+3b+3=0 \\ 4b(b+1)+3(b+1)=0 \\ (4b+3)(b+1)=0 \end{gathered}$$

Step 3. Use the Zero Product Property

The equations we get are

$$\begin{gathered} 4b+3=0 \\ b+1=0 \end{gathered}$$

Thus,

$$\begin{gathered} 4b+3=0 \\ b=\frac{-3}{4} \end{gathered}$$ or $$\begin{gathered} b+1=0 \\ b=-1 \end{gathered}$$

Step 4. Verify the answers

To verify the answer we will substitute the solutions separately into the original equation.

Put $b=\frac{-3}{4}$in the given equation,

$$\begin{gathered} 4{\left(\frac{-3}{4}\right)}^2+7\left(\frac{-3}{4}\right)=-3 \\ \frac{9}{4}-\frac{21}{4}=-3 \\ -3=-3 \end{gathered}$$

It is true.

Now, put $b=-1$in the given equation

$$\begin{gathered} 4{(-1)}^2+7(-1)=-3 \\ 4-7=-3 \\ -3=-3 \end{gathered}$$

It is also true.

Hence, the solutions are $b=\frac{-3}{4},-1.$