Question

How do you solve a quadratic equation without using a formula?

Answer 1

`"Sure there are"`

Explanation:

`"You could use"`
`"1) Completing the square :"`
`"e.g. x² - 4 x + 3 = 0"`
`=> "(x - 2)² - 1 = 0"`
`=> "(x - 2)² = 1"`
`=> "x - 2 = "pm"1"`
`=> "x = 1 or 3"`
`"2) Substituting x = y + p"`
`"e.g. x² + 5x + 6 = 0"`
`"x = y+p"`
`=> "(y+p)² + 5(y+p) + 6 = 0"`
`=> "y² + (2p+5) y + p²+5p+6 = 0"`
`"Now choose p so that 2p+5 = 0"`
`"So choose p = -5/2"`
`=> "y² - 25/4 + 6 = 0"`
`=> "y² = 1/4"`
`=> "y = "pm"1/2"`
`=> "x"+5/2 " = "pm" 1/2`
`=> "x = -2 or -3"`

Answer 2

See below.

Explanation:

There are different ways, but it will be dependent on the particular quadratic.

• **Completing the square **

Works for all quadratics

• Quadratic formula

Works for all quadratics

• Factoring

Can be difficult sometimes to find factors without knowing roots

• Graphically

Results from graphs are generally not very accurate.

• Iteration

This requires knowing roughly where the roots lie, in order to get convergence. A common iterative process is the Newton Raphson method.

So there other methods, but they are not as convenient as a formula.

Answer 3

How about geometrically...

Explanation:

No formula?

How about...

This solves the quadratic equation:

`x^2-a = 0`

To construct the square root of a positive real number geometrically, construct a semicircular arc with diameter `1+a`. Then the circle intercepts a perpendicular raised from the point on the base `1` unit along from one end at a distance `sqrt(a)`

With a bit more work, you can find real solutions of any quadratic geometrically.