Question #4bcdc

1 Answer

$\left(x + 21\right) \left(x - 29\right) = 0$

$x = - 21 , 29$

Explanation:

For parabolic equations, finding solutions to the equations can be done by two ways:

(1) Finding multiplicative factors of the constant or the rightmost term (i.e. $- 609$) that add up to the numerical value of the middle term ($- 8$). This method is recommended if the rightmost term doesn't have much factors. This would be difficult if the rightmost term were rational with several possible factors. Furthermore, this method requires a bit of a trail-and-error.

After multiple attempts of trial-and-error, I have found that

$\left(x + 21\right) \left(x - 29\right) = 0$

Wherein $609$ is the product of factors $- 29$ and $21$ and consequently the sum of these two factors is $- 8$.

(2) The quadratic equation.

The quadratic equation is $x = \frac{- b \pm \setminus \sqrt{{b}^{2} - 4 a c}}{2 a}$.

So what are the variables $a , b ,$ and $c$? These are just the numerical values of each term in the quadratic equation.

Specifically;
$a$ corresponds to the numerical value of the leftmost term;
$b$ corresponds to the numerical value of the middle term; and
$c$ corresponds to the numerical value of the rightmost term.

From the original equation,
$a = 1$
$b = - 8$
$c = - 609$

Then, plug-in these values into the quadratic equation:

(i) $x = \frac{- \left(- 8\right) + \setminus \sqrt{{\left(- 8\right)}^{2} - \left(4\right) \left(1\right) \left(- 609\right)}}{2 \left(1\right)} = 29$
(ii) $x = \frac{- \left(- 8\right) - \setminus \sqrt{{\left(- 8\right)}^{2} - \left(4\right) \left(1\right) \left(- 609\right)}}{21} = - 21$

Therefore,
From the obtained $x$ values, the factors should
$\left(x + 21\right) \left(x - 29\right) = 0$
to maintain the equality of the right side of the equation, $0$.