Question

For what values of x, the expression $15+4x-3x^{2}$ is negative?

Answer 1

As the Final Value of the given expression must be negative after the substitution of values of x in the expression we can say that :

15 + 4x - 3x² must be less than zero

⇒ 15 + 4x - 3x² < 0

⇒ 3x² - 4x - 15 > 0

⇒ 3x² - 9x + 5x - 15 > 0

⇒ 3x(x - 3) + 5(x - 3) > 0

⇒ (x - 3) (3x + 5) > 0

⇒ x > 3 and x < -5/3

The values of x for which the expression 15 + 4x - 3x² has negative values is

(-∞ , -5/3) ∪ (3 , ∞)

Answer 2

$\large{\sf{Solution -}}$

The given expression is-

$\large{\sf{15 + 4x - {3x}^{2} }}$

As the values of x should make it negative then we know that the value of x will be smaller than 0.

$\large{\sf{15 + 4x + {3x}^{2} < 0}}$

We can rearrange the terms of expression to form a general form of the expression.

We know that the general form of quadratic(as the highest power of the expression is 2 )equation is-

$\large{\sf{a {x}^{2} + bx + c}}$

Here,

• a, b and c are constant values of the quadratic equation.
• x is the only variable in the expression

To write the expression in general form we just have to rearrange the terms-

$\large{\sf{ {3x}^{2} + 4x + 15 < 0}}$

On splitting the middle terms we get-

$\large{\sf{ {3x}^{2} - 5x + 9x + 15 < 0}}$

$\large{\sf{ - x(5 + 3x) + 3(5 + 3x) < 0}}$

$\large{\sf{3(5 + 3x) - x(5 + 3x) < 0}}$

$\large{\sf{(3 - x)(5 + 3x) < 0}}$

Multiply both sides with -1

$\large{\sf{(x - 3)(3x + 5) < 0}}$

Now on using wavy curve method-

$\large{\sf{x \in ( - ∞, \frac{-5}{3}) ∪ (3,∞)}}$