Exercise 40 Page 211

Check if the related quadratic equation has real solutions or not.

{all real numbers }

Practice makes perfect

To solve the quadratic inequality algebraically, we will follow three steps.

Solve the related quadratic equation.
Plot the solutions on a number line if they exist.
Test a value to see if it satisfies the original inequality.

Step 1

We will start by solving the related equation. 3x^2+x=- 3 ⇔ 3x^2+ 1x+ 3=0We see above that a= 3, b= 1, and c= 3. Let's substitute these values into the Quadratic Formula to solve the equation.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- 1±sqrt(1^2-4( 3)( 3))/2( 3)

▼

Simplify right-hand side

BaseOne

1^a=1

x=- 1±sqrt(1-4(3)(3))/2(3)

Multiply

x=- 1±sqrt(1-36)/6

SubTerm

Subtract term

x=- 1±sqrt(- 35)/6

We have found a negative discriminant, - 35. Therefore, the related equation has no real solutions.

Step 2

Since the related equation does not have any real solution, we do not have any point to plot on a number line.We have two cases: either all x-values satisfy the original inequality or no x-values satisfy it.

Step 3

Finally, we must test a value to see if it satisfies the original inequality. Testing one value will help us to determine the solution set. For simplicity, we will choose x=0.

3x^2+x ≥ - 3

Substitute

x= 0

3( 0)^2+ 0? ≥- 3

▼

Simplify left-hand side

CalcPow

Calculate power

3(0)+0? ≥- 3

ZeroPropMult

Zero Property of Multiplication

0+0 ? ≥- 3