Question

How do you solve `7x^2 - 54 = 0`?

Answer 1

Although this is a quadratic equation, it is an easy one because there is no term in `x`. First, change the equation into the form `x² = c`. Then find both the positive and negative square root.

Explanation:

Divide both sides by 7 first to give: `x² - 54/7 = 0`

`x² = 54/7`
`x= +-sqrt(54/7)`

`x = 2.778 or x = -2.778` (3dp)
It could be given in surd form as well.

Remember that there are always two answers to a quadratic equation.

A longer method would be to use the quadratic formula with `a = 7, b=0 and c= -54`

Answer 2

`x=(3sqrt 6)/sqrt 7`

`x=-(3sqrt 6)/sqrt 7`

Explanation:

`7x^2-54=0`

Add `54` to both sides of the equation.

`7x^2=54`

Divide both sides by `7`.

`x^2=54/7`

Take the square root of both sides.

`x=+-sqrt(54/7)`

Simplify.

`x=+-(sqrt54)/(sqrt 7)`

Determine the prime factors for `54`.

`x=+-sqrt (2xx3xxcolor(blue)(3xx3))/(sqrt7)`

`x=+-sqrt((2xx3xxcolor(blue)(3^2)))/(sqrt7)`

Apply square root rule `sqrt(a^2)=a`.

`x=+-color(blue)(3color(black)sqrt(2xx3))/(sqrt 7)`

`x+-color(blue)(3color(black)sqrt 6)/(sqrt 7)`

Solution

`x=(3sqrt 6)/sqrt 7`

`x=-(3sqrt 6)/sqrt 7`