Question

use the quadratic formula to solve. express your answer in simplest form. 15b² - 8b - 3 = -4

Explanation:

Step1: Rewrite to standard quadratic form

$15b^2 - 8b - 3 + 4 = 0 \implies 15b^2 - 8b + 1 = 0$

Step2: Identify coefficients

For $ax^2+bx+c=0$, $a=15$, $b=-8$, $c=1$

Step3: Substitute into quadratic formula

The quadratic formula is $b=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:

$$ b=\frac{-(-8)\pm\sqrt{(-8)^2-4(15)(1)}}{2(15)} $$

Step4: Simplify discriminant and numerator

Calculate discriminant: $\sqrt{64-60}=\sqrt{4}=2$
Simplify numerator: $8\pm2$

Step5: Compute two solutions

First solution: $b=\frac{8+2}{30}=\frac{10}{30}=\frac{1}{3}$
Second solution: $b=\frac{8-2}{30}=\frac{6}{30}=\frac{1}{5}$

Answer:

$b=\frac{1}{3}$ and $b=\frac{1}{5}$