How do you find the discriminant for $9x^2+24x=-16$ and determine the number and type of solutions?

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Answer 1 · 2018-04-12T01:21:13

See a solution process below:

First, write the equation in standard form:

$9x^2 + 24x + color(red)(16) = -16 + color(red)(16)$

$9x^2 + 24x + 16 = 0$

For $ax^2 + bx + c = 0$ , the values of $x$ which are the solutions to the equation are given by:

$x = (-b +- sqrt(b^2 - 4ac))/(2a)$

The discriminate is the portion of the quadratic equation within the radical: $color(blue)(b)^2 - 4color(red)(a)color(green)(c)$

If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions

To find the discriminant for this problem substitute:

$color(red)(9)$ for $color(red)(a)$

$color(blue)(24)$ for $color(blue)(b)$

$color(green)(16)$ for $color(green)(c)$

$color(blue)(24)^2 - (4 * color(red)(9) * color(green)(16))$

$576 - (36 * color(green)(16))$

$576 - 576$

$0$

Because the discriminate is $0$ there is just one solution to this problem.

How do you find the discriminant for 9x^2+24x=-169x^2+24x=-16 and determine the number and type of solutions?