Exercise 50 Page 287

Solve the related quadratic equation, plot the solutions on a number line, and test a value from each interval.

{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 exist.
Test a value to see if it satisfies the original inequality.

Step 1

We will start by solving the related equation. 5x^2+ 1x+ 3=0We see above that a= 5, 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( 5)( 3))/2( 5)

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Simplify right-hand side

BaseOne

1^a=1

x=- 1±sqrt(1-4(5)(3))/2(5)

Multiply

x=- 1±sqrt(1-60)/10

SubTerm

Subtract term

x=- 1±sqrt(- 59)/10

We have found a negative discriminant, - 59. 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-value satisfies 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.

5x^2+x+3 ≥ 0

Substitute

x= 0

5( 0)^2+ 0+3? ≥ 0

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Simplify left-hand side

CalcPow

Calculate power

5(0)+0+3? ≥ 0

ZeroPropMult

Zero Property of Multiplication

0+0+3? ≥ 0