# Find Turning Point by Completing the Square

In this revision note, you will learn the steps to find turning point by completing the square of Quadratic

Graphs.

## 3 Forms of Quadratic Expressions

A quadratic expression can be written in 3 main forms:

(i) General form: *ax ^{2} + bx + c*

(ii) Factorised form: *(x + p)(x +q)*

(iii) “Completed square” form: *a(x + h) ^{2} + k*

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## Completing the Square

Apply the following steps to “complete the square”.

Observe the general form. If the coefficient of factorise the coefficient of xfor the terms with ^{2 }xand ^{2 }x only.
| Example 1
2 x – 1 = 2( x8^{2} + x ) – 1 |

(a + b)^{2 }– b^{2}With reference to Example 1: · x^{2},· · 2 · **In general, | 2 x – 1 = 2( x8^{2} + x ) – 1= 2[( This is the “completed square”. |

Simplify the constants to obtain the “completed square” form: | 2 x – 1 = 2( x8^{2} + x ) – 1= 2[( = 2( = 2( |

## Find the Turning Point by Completing the Square

### Completing the Square Example 1

State the minimum value of = x^{2} − 4x + 10 and the corresponding value of *x.*

Solution

= *x*^{2} − 4*x* + 10

= (*x* − 2)^{ 2} − (−2)^{ 2} + 10

= (*x*− 2)^{ 2} + 6

Since ( *x* − 2) ^{2} ≥ 0 for all real values of x, then ( *x* − 2)^{ 2} + 6 ≥ 6 for all real values of x.

The minimum value of y is 6, when *x* = 2. (ans)

*Tip: To find the turning point from the “completed square” form, we let the **expression inside the bracket be zero**.

Note: Since the graph is a U-shaped graph, the turning point is a minimum point and the coordinates are (2 , 6).

**Completing the Square Example 2**

State the maximum value of = −2*x* ^{2} − 10*x* + 5 and the corresponding value of *x*.

Solution

y = −2*x*^{2} − 10*x* + 5

= −2 ( *x* + 5/2 )^{2} + 35/2

Since ( *x* + 5/2 )^{2} ≥ 0 for all real values of *x*, then −2 ( *x* + 5/2 )^{2} ≤ 0 for all real values of x.

And −2 ( *x* + 5/2 )^{2} + 35/2 ≤ 35/2

The maximum value of y is 35/2 , when *x* = − 5/2 . (ans)

Note: Since the graph is an inverted-U graph, the turning point is a maximum point and the coordinates are (− 5/2 , 35/2 ).

## Summary of how to find a turning point by completing the square

In general, the coordinates of the turning point of a quadratic graph after completing the square, y = *a(x + h)*^{2} + k is **always** given by ( −h, k). This information is very useful for graph sketching.

## Last Minute Revision for O Level Math?

## Check out our exam guide on other topics here!

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