2.3a Forming Quadratic Equations from Given Roots

Posted on April 18, 2020 by user

2.5 Forming Quadratic Equations from Given Roots

Let α and β be the roots of the equation ax²+ bx + c = 0, this means

Example:

Find the quadratic equation from the respective roots below:
(a) 3, −1

(b) −2, ¼

Solution:

2.2.2c Solving Quadratic Equations – Quadratic Formula

Posted on April 18, 2020 by user

The quadratic equation

a x^{2} + b x + c = 0

can be solved by using the quadratic formula

\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Example
Use the quadratic formula to find the solutions of the following equations.
a.

x^{2} + 5 x - 24 = 0

x (x + 4) = 10

Answer
(a)
For the equation

x^{2} + 5 x - 24 = 0

a = 1, b = 5, c = -24

\begin{array}{l} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ x = \frac{- (5) \pm \sqrt{{(5)}^{2} - 4 (1) (- 24)}}{2 (1)} \\ x = \frac{- 5 \pm \sqrt{121}}{2} \\ x = - 8 or x = 3 \end{array}

(b)

\begin{array}{l} x (x + 4) = 10 \\ x^{2} + 4 x - 10 = 0 \\ a = 1, b = 4, c = - 10 \\ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ x = \frac{- (4) \pm \sqrt{{(4)}^{2} - 4 (1) (- 10)}}{2 (1)} \\ x = \frac{- 4 \pm \sqrt{56}}{2} \\ x = 1.742 or x = - 5.742 \end{array}

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}

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

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