Question

Solve:

$45b^2-2b=-17b$

Short answer

The solutions of the quadratic equations $45b^2-2b=-17b$are $b=-\frac{1}{3},0$

Step-by-step solution

Step 1. Given Information

An equation:

$45b^2-2b=-17b$

Step 2. Write the quadratic equation in standard form.

$$\begin{gathered} 45b^2-2b=-17b \\ 45b^2-2b+17b=0 \\ 45b^2+15b=0 \end{gathered}$$

Step 3. Factor the quadratic expression

Factorize the equation,

$$\begin{gathered} 45b^2+15b=0 \\ 15b(3b+1)=0 \end{gathered}$$

Step 4. Use the Zero-Product property

Use the Zero Product Property and set each factor to zero.

$$\begin{gathered} 15b=0 \\ b=0 \\ 3b+1=0 \\ 3b=-1 \\ b=-\frac{1}{3} \end{gathered}$$

The value of b is $b=-\frac{1}{3},0$

Step 5. Check the answer

Check by substituting the values in the original equation,

When $b=-\frac{1}{3}$,

$$\begin{gathered} 45{(-\frac{1}{3})}^2-2(-\frac{1}{3})=-17(-\frac{1}{3}) \\ \frac{15}{3}+\frac{2}{3}=\frac{17}{3} \\ \frac{17}{3}=\frac{17}{3} \end{gathered}$$

When $b=0$,

$$\begin{gathered} 45{\left(0\right)}^2-2\left(0\right)=-17\left(0\right) \\ 0=0 \end{gathered}$$

Hence the equations hold true.