Question

Complete the square to write the quadratic expression in vertex form.

Answer 1

Answer:

1. (x - 3)² = 8

2. (x + 2)² = 3

3. (x + 6)² = $\frac{101}{2}$

4. (x + 3)² = 27

5. (x + 4)² = 13

6.  $\bigg( x - \frac{15}{9} \bigg) ^2 = \frac{261}{81} = \frac{29}{9}$

Step-by-step explanation:

Completion of Square: $(x - a) ^2 = x^2 - 2ax + a^2$

In the following problems the terms in the RHS of the above equation may be missing. We balance the equation. Simplify it and re write it in terms of LHS.

1. x² - 6x + 1 = 0

Taking the constant term to the other side, we get:

x² - 6x = - 1

⇒ x² - 2(3)x = -1

⇒ x² -2(3)x + 9 = - 1 + 9  [Adding 9 to both the sides]

⇒ x² -2(3)x + 3² = 8

⇒ (x - 3)² = 8 is the answer.

2. 3x² + 12x + 3 = 0

Note that the co-effecient of x² is not 1. We make it 1, by dividing the whole equation by 3. And then proceed like the previous problem.

3x² + 12x = -3

Dividing by 3 through out, x² + 4x = - 1

⇒ x² + 2(2) + 4 = -1 + 4

⇒ x² +2(2) + 2² = 3

⇒ (x + 2)² = 3 is the answer.

3. 2x² + 24x = 29

x² + 12x = $\frac{29}{2}$

⇒ x² + 2(6)x + 36 = $\frac{29}{2}$ + 36

⇒ x² + 2(6)x + 6² = $\frac{29 + 72}{2}$

⇒ (x + 6)² = $\frac{101}{2}$ is the answer.

4. x² + 6x - 18 = 0

x² + 6x = 18

⇒ x² + 2(3)x = 18

⇒ x² + 2(3)x + 9 = 18 + 9

⇒ x² + 2(3)x + 3² = 27

⇒ (x + 3)² = 27 is the answer.

5. x² + 8x + 3 = 0

x² + 8x = -3

⇒ x² + 2(4)x = -3

⇒ x² + 2(4)x + 16 = - 3 + 16

⇒ x² + 2(4)x + 16 = 13

⇒ (x + 4)² = 13 is the answer.

6. 9x² - 30x + 6 = 0

9x² - 30x = - 6

⇒ x² $- \frac{30}{9}$ x = - 6

$\implies x^2 -2 \bigg( \frac{15}{9} \bigg )x + \frac{225}{81} = - 6 + \frac{225}{81}$

$\implies x^2 - 2\bigg( \frac{15}{9} \bigg ) x + \bigg ( \frac{15}{9} \bigg ) ^2 = \frac{261}{81}$

$\bigg( x - \frac{15}{9} \bigg) ^2 = \frac{261}{81} = \frac{29}{9}$ is the answer.