Exercise 4 Page 194

To solve the given equation by factoring, we will start by factoring out the greatest common factor (GCF). We have rewritten the left-hand side as a product of two factors. Now, we will apply the Zero Product Property to solve the equation. From Equation (I), we found that one solution is a=0. To find other solutions, we will solve Equation (II). Note that this is a quadratic equation. Thus, we will use the Quadratic Formula. ax^2+bx+c=0 ⇔ x=- b±sqrt(b^2-4ac)/2a Let's rewrite Equation (II) in terms of the x-variable, so that our variable is not confused with the a-variable from the formula. a^2-4a+4=0 ⇔ x^2-4x+4=0 At first we need to identify a, b, and c. x^2-4x+4=0 ⇔ 1x^2+( -4)x+( 4)=0 We see that a= 1, b= -4, and c= 4. Let's substitute these values into the formula and solve for x.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- ( -4)±sqrt(( -4)^2-4( 1)( 4))/2( 1)

▼

Solve for x

NegNeg

- (- a)=a

x=4±sqrt((- 4)^2-4(1)(4))/2(1)

CalcPow

Calculate power

x=4±sqrt(16-4(1)(4))/2(1)

IdPropMult

Identity Property of Multiplication

x=4±sqrt(16-4(4))/2

Multiply

Multiply

x=4±sqrt(16-16)/2

SubTerms

Subtract terms

x=4±sqrt(0)/2

CalcRoot

Calculate root

x=4± 0/2

Since adding or subtracting zero does not change the value of a number, the numerator will simplify to 4. Therefore, we will get only one value of x. x= 4/2 ⇔ x=2 Using the Quadratic Formula, we found that the solution of the quadratic equation is x=2, which is a double root. We can now go back to the given equation. a^3-4a^2+4a=0 We found in total two solutions for a. a=2 and a=0