Exercise 29 Page 195

Practice makes perfect

a In the Quadratic Formula, b^2-4ac is the discriminant.

ax^2+bx+c=0 ⇔ x=- b±sqrt(b^2-4ac)/2a Let's first rewrite the given equation in standard form.

- 3x^2-7x+2=6

LHS-6=RHS-6

- 3x^2-7x-4=0

Having rewritten the equation, we can identify the values of a, b, and c. - 3x^2-7x+2=6 ⇔ - 3x^2+( - 7)x+( - 4)=0 Now, let's evaluate the discriminant.

b^2-4ac

SubstituteValues

Substitute values

( - 7)^2-4( - 3)( - 4)

▼

Simplify

NegBaseToPosBase

(- a)^2=a^2

49-4(- 3)(- 4)

MultNegNegOnePar

- a(- b)=a* b

49+12(- 4)

MultPosNeg

a(- b)=- a * b

49-48

SubTerm

Subtract term

The discriminant is 1.

b We want to use the discriminant of the given quadratic equation to determine the number and type of the roots. If we do not want to know the exact values of the roots, we only need to work with the discriminant. From Part A, we know that the discriminant of the given equation is 1.

Equation:& - 3x^2-7x+2=6 Discriminant:& 1 Since the discriminant is greater than zero and a perfect square, the quadratic equation has two rational roots.

c We will use the Quadratic Formula to find the exact solutions of the given equation.

x=- b±sqrt(b^2-4 a c)/2 a Recall that we have already identified the values of a, b, and c in Part A, as well as the discriminant, b^2-4ac. a= - 3, b= - 7, c= - 4 Discriminant: 1 Let's substitute these values into the Quadratic Formula.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=-( - 7)±sqrt(1)/2( - 3)

▼

Simplify

NegNeg

- (- a)=a

x=7±sqrt(1)/2(- 3)

CalcRoot

Calculate root

x=7± 1/2(- 3)