BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.5, Problem 2E

(a)

To determine

To explain: The method of factoring to solve the equation x24x5=0.

Expert Solution

Explanation of Solution

Property used:

Zero-product property:

If ab=0, then a=0orb=0, or both.

Calculation:

Note that, the expression on the left side of a quadratic equation in the standard form ax2+bx+c=0,a0 can be factored into the product of two first-degree polynomials.

Then, by using the zero-product property set each factor equal to zero. Then, the resulting equations are solved to obtain the given quadratic equation.

Thus, the given equation can be solved by factoring the expression x24x5 as follows.

x24x5=0x25x+x5=0x(x5)+1(x5)=0(x+1)(x5)=0

Now, use the zero-product property to find the solutions.

(x+1)(x5)=0x+1=0orx5=0[Zeroproductproperty]x=1andx=5

Thus, the solutions of equation x24x5=0 by the method of factoring  are x=1 and x=5.

(b)

To determine

To explain: The method of completing the square to solve the equation x24x5=0.

Expert Solution

Explanation of Solution

Method used:

Completing the square:

1. Begin by writing the equation with the constant on the right side.

2. If the coefficient of x2 is 1, then complete the square on the left side by adding the square of the half of the coefficient of x and balance it by adding the same value on the right side.

3. If the coefficient of x2 is not 1, then divide all the terms by the coefficient and complete the square.

That is, to make x2+bx a perfect square, add (b2)2, the square of half the coefficient of x. This gives the perfect square x2+bx+(b2)2=(x+b2)2.

Calculation:

Consider the given equation x24x5=0.

Add 5 on both the sides of equation x24x5=0.

x24x5=0x24x=5

Here, the coefficient of x2 is 1. Therefore, complete the square on the left side by adding the square of the half of the coefficient of x and balance it by adding the same value on the right side

That is, add (42)2 to both sides and simplify.

x24x+(2)2=4+(2)2[Add(42)2=(2)2]x24x+4=5+4(x2)2=9[Perfectsquare](x2)=±3[Takesquareroot]

Add 2 on both sides of the equation to obtain the final solution.

(x2)=±3x=2±3[Add2onbothsides]x=2+3orx=23x=5orx=1

Thus, the solution of equation x24x5=0 by the method of completing the square are x=1 and x=5.

(c)

To determine

To explain: The method of using quadratic formula to solve the equation x24x5=0.

Expert Solution

Explanation of Solution

Formula used:

Quadratic formula:

The solution of a quadratic equation of the form ax2+bx+c=0,a0 can be obtained by using the quadratic formula x=b±b24ac2a.

Calculation:

Consider the given equation x24x5=0.

Compare this equation with the general form ax2+bx+c=0.

Here a=1,b=4and c=5.

Substitute 1 for a, 4 for b and 5 for c in the quadratic formula to find the solution.

x=(4)±(4)24(1)(5)21=4±16(20)2=4±362=4±62

Simplify further as follows.

x=4±62x=2±3x=2+3orx=23x=5orx=1

Thus, the solution of equation x24x5=0 by using quadratic formula are x=1 and x=5.

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