# How do you solve 23p = 5p² + 24?

Jul 22, 2016

$p = 3 , p = \frac{8}{5}$

#### Explanation:

Solve quadratic equations by making them equal to 0.

$5 {p}^{2} - 23 p + 24 = 0$

Factorise: Find factors of 5 and 8 which add to give 23. Signs will be the same they are both negative.

Cross multiply the factors:

$\text{5 8} \Rightarrow 1 \times 8 = 8$
$\text{1 3" rArr 5 xx3 = 15" } 15 + 8 = 23$

$\left(5 p - 8\right) \left(p - 3\right) = 0 \text{ we have two factors}$

Either of the factors could be equal to 0.

$5 p - 8 = 0 , \text{ or } p - 3 = 0$
$5 p = 8 \text{ } p = 3$
$p = \frac{8}{5}$

Jul 22, 2016

$\frac{8}{5} \mathmr{and} 3$

#### Explanation:

$y = 5 {p}^{2} - 23 p + 24.$
Solve this quadratic equation by the new Transforming Method (Socratic Search).
Method. First find the 2 real roots of the transformed equation,
y' = p^2 - 23p + 120 = 0. Next, divide the answers by (a) to get the
2 real roots of y.
Two real roots of y' have same sign (Rule of signs)
Factor pairs of (ac = 120) --> (4, 30)(5, 24)(8, 15). This last sum is
23 = -b. The 2 real roots of y' are: 8 and 15.
Back to original equation y, the 2 real roots are: $p 1 = \frac{8}{a} = \frac{8}{5} = \frac{8}{5}$, and $p 2 = \frac{15}{a} = \frac{15}{5} = 3$