Introduction
- Completing the squares is a powerful technique used in algebra to solve quadratic equations.
- It is important to study the Completing the Square because it helps solve quadratic equations, understand the shape of parabolas, find the turning point etc.
Watch: Completing the Square
What is Completing the Square?
- Completing the square is an algebraic technique that transforms a quadratic expression into the sum of a perfect square binomial and a remaining constant term.
$$ax^2 + bx + c \longrightarrow a(x + d)^2 + e$$
- This method reveals key properties of quadratic functions.
Steps to Solve Completing the Square
- Completing the square is a method used to rewrite quadratic expressions in the form:
$$a(x – h)^2 + k$$
- Steps to convert Quadratic expression into the form of $a(x – h)^2 + k$ by Completing the Square:
1
Identify the Quadratic expression $ax^2 + bx + c$.
2
Solve for $\frac{b}{2a}$.
3
Re-write the equation as,
$$a\left(x + \frac{b}{2a}\right)^2 + k$$
4
Plug $x = -\frac{b}{2a}$ into your original equation and solve for $k$.
5
Simplify it.
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Book a Consultation →Finding the Turning Point
- We can also find the Turning point by Completing the Square
$$a\left(x + \frac{b}{2a}\right)^2 + k$$
$$y = a(x – h)^2 + k$$
- $h$ and $k$ are the turning point (vertex) of the parabola.
- $a$ determines the parabola’s width and direction (upwards if $a > 0$, downwards if $a < 0$).
Solved Examples
Solved Example
Factorize $2x^2 + 6x + 9$ by completing the square.
SOLUTION
1
Identify the Quadratic expression. Here:
- $a = 2$ (coefficient of $x^2$)
- $b = 6$ (coefficient of $x$)
- $c = 9$ (constant term)
2
Solve for:
$$\frac{b}{2a} = \frac{6}{2(2)} = \frac{3}{2}$$
3
Re-write the equation as,
$$a\left(x + \frac{b}{2a}\right)^2 + k$$
$$2\left(x + \frac{3}{2}\right)^2 + k$$
4
Plug $x = \frac{-b}{2a}$ to your original equation and solve for $k$.
$$2\left(\frac{-3}{2}\right)^2 + 6\left(\frac{-3}{2}\right) + 9$$
$$2\left(\frac{9}{4}\right) – \frac{18}{2} + 9 = \frac{9}{2}$$
5
Simplify it.
$$2\left(x + \frac{3}{2}\right)^2 + \frac{9}{2}$$
Final Answer: $2\left(x + \frac{3}{2}\right)^2 + \frac{9}{2}$
Solved Example
Find the turning point of $y = 2(x – 2)^2 – 3$
SOLUTION
Identify the Turning Point:
$y = 2(x – 2)^2 – 3$
- $h = 2$ (the x-coordinate of the vertex).
- $k = -3$ (the y-coordinate of the vertex).
Final Answer: The Turning point (vertex) of the parabola is $(2, -3)$.
Solved Example
Factorize $x^2 + 6x + 5$ by completing the square.
SOLUTION
1
Identify the Quadratic expression. Here:
- $a = 1$ (coefficient of $x^2$)
- $b = 6$ (coefficient of $x$)
- $c = 5$ (constant term)
2
Solve for:
$$\frac{b}{2a} = \frac{6}{2(1)} = 3$$
3
Re-write the equation as,
$$a\left(x + \frac{b}{2a}\right)^2 + k$$
$$(x + 3)^2 + k$$
4
Plug $x = \frac{-b}{2a}$ to your original equation and solve for $k$.
$$(-3)^2 + 6(-3) + 5$$
$$9 – 18 + 5 = -4$$
5
Simplify it.
$$(x + 3)^2 – 4$$
Final Answer: $(x + 3)^2 – 4$
Solved Example
Find the Turning Point of $y = 3\left(x – \frac{5}{3}\right)^2 – \frac{4}{3}$
SOLUTION
Identify the Turning Point:
$y = 3\left(x – \frac{5}{3}\right)^2 – \frac{4}{3}$
Where,
- $h = \frac{-5}{3}$ (the x-coordinate of the vertex).
- $k = \frac{-4}{3}$ (the y-coordinate of the vertex).
Final Answer: The Turning Point of parabola is $\left(\frac{-5}{3}, \frac{-4}{3}\right)$
Solved Example
Given the quadratic equation:
$y = 2x^2 + 8x + 5$
Rewrite the equation in completed square form and hence, find the turning point of the parabola.
SOLUTION
1
Identify the Quadratic expression. Here:
- $a = 2$ (coefficient of $x^2$)
- $b = 8$ (coefficient of $x$)
- $c = 5$ (constant term)
2
Solve for:
$$\frac{b}{2a} = \frac{8}{2(2)} = 2$$
3
Re-write the equation as,
$$a\left(x + \frac{b}{2a}\right)^2 + k$$
$$(x + 2)^2 + k$$
4
Plug $x = \frac{-b}{2a}$ to your original equation and solve for $k$.
$$2(-2)^2 + 8(-2) + 5$$
$$8 – 16 + 5 = -3$$
5
Simplify it.
$$2(x + 2)^2 – 3$$
6
Identify the Turning Point:
- Vertex form:
$$a(x – h)^2 + k$$
Comparing with it, $h = -2$ and $k = -3$
Final Answer: The Turning Point is $(-2, -3)$.
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