Exercise 50 Page 332

Find the roots and use them to graph the related function.

{ x| x ≤ - sqrt(21) + 6 or x ≥ sqrt(21) + 6 }

Practice makes perfect

We will start by simplifying the inequality a bit. Then we will sketch the related quadratic function.

Simplifying the Inequality

Note that both sides of the inequality consists of the constant terms. Let's simplify it then!

2(x-6)^2-5 ≥ 37

SubIneq

LHS-37≥RHS-37

2(x-6)^2-42 ≥ 0

FactorOut

Factor out 2

2((x-6)^2-21) ≥ 0

DivIneq

.LHS /2.≥.RHS /2.

(x-6)^2-21 ≥ 0

We will now find zeros of the related quadratic function.

y = (x-6)^2 - 21

Finding the Roots

Finally, to find the roots of the quadratic function, we will solve the equation by taking the square root.

(x-6)^2 - 21=0

AddEqn

LHS+21=RHS+21

(x-6)^2=21

SqrtEqn

sqrt(LHS)=sqrt(RHS)

x-6=± sqrt(21)

AddEqn

LHS+6=RHS+6

x=± sqrt(21) + 6

We will now find the first and second solutions by using the positive and negative signs.

x=± sqrt(21) + 6
x=sqrt(21) + 6	x=-sqrt(21) + 6
x≈ 10.583	x≈1.417

Solving the Inequality

To solve the inequality, we must find the leading coefficient of the quadratic function.

(x-6)^2-21

ExpandNegPerfectSquare

(a-b)^2=a^2-2ab+b^2

x^2-2*6x+6^2-21

Multiply

x^2-12x+6^2-21

CalcPow

Calculate power

x^2-12x+36-21

SubTerm

Subtract term

x^2-12x+15

We can now identify the value of a. x^2-12x+15 ⇔ 1x^2-12x+15 Therefore, the graph of the quadratic function opens upward, since a= 1 is greater than zero. The solution of the given quadratic inequality, ax^2+bx+c≥0, consists of x-values for which the graph of the related quadratic function lies on and above the x-axis.

We see that the graph lies above the x-axis at x ≤ - sqrt(21) + 6 and x ≥ sqrt(21) + 6. { x| x ≤ - sqrt(21) + 6 or x ≥ sqrt(21) + 6 }

Subchapter links

Communicate Your Answer p.327

Monitoring Progress p.328-330

Exercises p.331-332

Exercise 50 Page 332

Hints

Check the answer

Practice exercises

Assess your skill

Simplifying the Inequality

Finding the Roots

Solving the Inequality