Question

Choose a method (factoring. square root property, completing the square, and the quadratic formula) to solve each of the following quadratic equations.

Each method can be used only ONCE

A) 4x2 – 27 = 0

B) 4x2 – 8x – 5 = 0

C) 4x2 – 8x – 12 = 0

D) 4x2 – 9x – 7 = 0

Answer 1

Part A)
`x=(+/-)(3√(3))/(2)`

Part B)
`x=2.5` and
`x=-0.5`

Part C)
`x=-1` and
`x=3`

Part D)
`x=(9+√(193))/(8)` and
`x=(9-√(193))/(8)`

Explanation:

Part A) we have

`4x^(2)-27=0`

we know that

The square root property states that if we have an equation with a perfect square on one side and a number on the other side, then we can take the square root of both sides and add a plus or minus sign to the side with the number and solve the equation.

isolate the term that contains the squared variable

`4x^(2)=27`

`x^(2) =(27)/(4)`

take the square root of both sides

`x=(+/-)(√(27))/(2)`

simplify

`x=(+/-)(3√(3))/(2)`

Part B) we have

`4x^(2)-8x-5=0`

Using the quadratic equation

The formula to solve a quadratic equation of the form

`ax^(2) +bx+c=0` is

`x=\frac{-b(+/-)\sqrt{b^(2)-4ac}} {2a}`

in this problem we have

`4x^(2)-8x-5=0`

so

`a=4\\b=-8\\c=-5`

substitute in the formula

`x=\frac{-(-8)(+/-)\sqrt{-8^(2)-4(4)(-5)}} {2(4)}`

`x=(8(+/-)√(144))/(8)`

`x=(8(+/-)12)/(8)`

`x=(8(+)12)/(8)=2.5`

`x=(8(-)12)/(8)=-0.5`

Part C) we have

`4x^(2)-8x-12=0`

Using Factoring

Simplify the expression first

Divide by 4 both sides

`x^(2)-2x-3=0`

Find two numbers a and b such that

a+b=-2

ab=-3

Solve the system by graphing

The solution is a=1, b=-3

see the attached figure

so

`4x^(2)-8x-12=4(x-1)(x+3)`

The solutions are

`x=1, x=-3`

Part D) we have

`4x^(2)-9x-7=0`

Solve by completing the square

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`4x^(2)-9x=7`

Factor the leading coefficient

`4(x^(2)-(9)/(4)x)=7`

Complete the square. Remember to balance the equation by adding the same constants to each side

`4(x^(2)-(9)/(4)x+(81)/(64))=7+(81)/(16)`

`4(x^(2)-(9)/(4)x+(81)/(64))=(193)/(16)`

Rewrite as perfect squares

`4(x-(9)/(8))^(2)=(193)/(16)`

`(x-(9)/(8))^(2)=(193)/(64)`

take square root both sides

`(x-(9)/(8))=(+/-)(√(193))/(8)`

`x=(9+√(193))/(8)`

`x=(9-√(193))/(8)`