QuestionFind the roots of $2 x = -5 + 3 x^{2}$ and the values of $c$ for which $3 x^{2} + 5 x + c > 0$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is x = -5 +3x^ . The quadratic equation is $3x^ +5x + c$
3. We need to find the nature of the roots of the given equation4. We also need to find the range of values of $c$ for which the quadratic equation is always positive

STEP 2

First, we need to rewrite the given equation in the standard form of a quadratic equation, $ax^2 + bx + c =0$ .
$x^2 -2x +5 =0$

STEP 3

The nature of the roots of a quadratic equation is determined by the discriminant, $b^2 -ac$ . If the discriminant is greater than0, the roots are real and distinct. If it is equal to0, the roots are real and equal. If it is less than0, the roots are complex.
$= b^2 -ac$

STEP 4

Now, plug in the values for $a$ , $b$ , and $c$ from the equation $3x^2 -2x + =0$ into the discriminant formula.
$= (-2)^2 -43$

STEP 5

Calculate the discriminant.
$=4 -60 = -56$

STEP 6

Since the discriminant is less than0, the roots of the equation are complex.

STEP 7

Now, let's find the range of values of $c$ for which the quadratic equation $3x^2 +5x + c$ is always positive.

STEP 8

For a quadratic equation $ax^2 + bx + c$ to be always positive, it must open upwards ( $a >0$ ) and its discriminant must be less than or equal to0.
$= b^2 -4ac \leq0$

STEP 9

Now, plug in the values for $a$ and $b$ from the equation $3x^2 +5x + c$ into the discriminant inequality.
$=5^2 -43c \leq$

STEP 10

implify the inequality.
$25 -12c \leq0$

STEP 11

olve the inequality for $c$ .
$c \geq \frac{25}{}$ So, the quadratic equation $3x^ +5x + c$ is always positive when $c \geq \frac{25}{}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms