2.2b Solving Quadratic Equations – Completing the Square (Examples)

Posted on April 18, 2020 by user

2.4.2a Solving Quadratic Equations – Completing the Square (Examples)

(A) Steps to solve quadratic equation using completing the square
• make sure that the coefficient of x is 1.
• Rewrite the equation ax²+ bx + c = 0 in the form ax²+ bx = –c.
• Add

{(\frac{coefficient of x}{2})}^{2}

${(\frac{coefficient of x}{2})}^{2}$ to both side of the equation.

Example:

Solve the following quadratic equations by completing the square.
(a) x² – 6x – 3 = 0

(b) 2x² – 5x – 7 = 0

Solution:

2.2b Solving Quadratic Equations – Completing the Square

Posted on April 18, 2020 by user

2.4.2 Solving Quadratic Equations – Completing the Square

(A) The Perfect Square

1. The expression x² + 2x + 1 can be written in the form (x + 1)², it is called a “perfect square”.

2. If the algebraic expression on the left hand side of the quadratic equation is a perfect, the roots can be easily obtained by finding the square roots.

Example:
Solve each of the following quadratic equation

(a) (x + 1)² = 25

(b) x² − 8x + 16 = 49

Solution:

(a)

(x + 1)² = 25

(x + 1)² = ±√25

x = −1 ± 5

x = 5 or x = −6

(b)

x² − 8x + 16 = 49

(x − 4)² = 49

(x − 4) = ±√49

x = 4 ± 7

x = 11 or x = −3

(B) Solving Quadratic Equation by Completing the Square

1. To solve quadratic equation, we make the left hand side of the equation a perfect square.

2. To make any quadratic expression x² + px into a perfect square, we add the term (p/2)² to the expression and this will make

3. The following shows the steps to solve the equation by using completing the square method for quadratic equation ax²+ bx = – c.

(a) Rewrite the equation in the form ax² + bx = – c.

(b) If the coefficient a ≠ 1, reduce it to 1 (by dividing).

(c) Add (p/2)² to both sides of the equation.

(d) Write the expression on the left hand side as a perfect square.

(e) Solve the equation.

Solving Quadratic Equations

Posted on April 18, 2020 by user

Solving Quadratic Equations

To solve a quadratic equation means to find all the roots of the quadratic equations.

Example:
Find the roots of the quadratic equations
a.

x^{2} = 9

$x^{2} = 9$
b.

2 x^{2} - 98 = 0

$2 x^{2} - 98 = 0$

Answer:
(a)

$\begin{array}{l} x^{2} = 9 \\ x = \pm \sqrt{9} \\ x = \pm 3 \end{array}$

(b)

$\begin{array}{l} 2 x^{2} - 98 = 0 \\ 2 x^{2} = 98 \\ x^{2} = \frac{98}{2} = 49 \\ x = \pm \sqrt{49} = \pm 7 \end{array}$

A quadratic equation may be solve by using one of the following method

Factorisation
Completing the square
Using quadratic formula

Roots of Quadratic Equation – Example

Posted on April 18, 2020 by user

Remember : Roots of a quadratic equation are the values of variables/unknowns that satisfy the equation.

Example
(a) Given x = 3 is the root of the quadratic equation

$x^{2} + 2 x + p = 0$ , find the value of p.
(b) The roots of the quadratic equation

$3 x^{2} + h x + k = 0$ are -2 and 4. Find the value of h and k.

General Form of Quadratic Equation

Posted on April 18, 2020 by user

General Form of Quadratic Equation

General form of a quadratic equation is

$a x^{2} + b x + c = 0$
where a, b, and c are constants and a≠0.

*Note that the highest power of an unknown of a quadratic equation is 2.

Example (Find the values of a, b and c)
Rewrite the following into the general form of a quadratic equation. Find the values of a, b, and c.
(a)

${(3 x - 5)}^{2} = 0$
(b) (x - 8)(x + 8) = 0

Solution

Quadratic Equations (Introduction)

Posted on April 18, 2020 by user

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree.
A quadratic equation has only one variable
The highest power of the variable is 2.

Example of Quadratic Equation
The followings are some examples of quadratic equations

$2 x^{2} + 3 x + 4 = 0$
$t^{2} = \frac{2}{5}$
$y (6 y - 3) = 5$
$a - 3 y + 2 y^{2} = 2$ where a is a constant.

Example of Non Quadratic Equation

$2 x + 1 = 0$ , (Reason: The highest power of x ≠ 2.)
$2 x^{3} + 1 = x$ , (Reason: The highest power of x ≠ 2.)
$t^{2} + \frac{5}{t} = 3$ , (Reason: The present of the term $\frac{5}{t}$ .)