Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Certainly! We will rewrite each given quadratic expression in the form [tex]\((ax + b)^2 + c\)[/tex]. This process is called completing the square. Let's go step-by-step for each expression.
### 1. [tex]\(x^2 + 6x - 10\)[/tex]
1. Identify the coefficient of [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[ a = 1, \quad b = 6 \][/tex]
2. Half the coefficient of [tex]\(x\)[/tex], then square it.
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
3. Add and subtract this square inside the expression:
[tex]\[ x^2 + 6x + 9 - 9 - 10 \][/tex]
[tex]\[ (x + 3)^2 - 19 \][/tex]
Thus, the expression is:
[tex]\[ (x + 3)^2 - 19 \][/tex]
### 2. [tex]\(x^2 - 3x + 5\)[/tex]
1. Identify the coefficient of [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[ a = 1, \quad b = -3 \][/tex]
2. Half the coefficient of [tex]\(x\)[/tex], then square it.
[tex]\[ \left(\frac{-3}{2}\right)^2 = \left(\frac{-3}{2}\right)^2 = \frac{9}{4} \][/tex]
3. Add and subtract this square inside the expression:
[tex]\[ x^2 - 3x + \frac{9}{4} - \frac{9}{4} + 5 \][/tex]
[tex]\[ \left(x - \frac{3}{2}\right)^2 + 5 - \frac{9}{4} \][/tex]
[tex]\[ \left(x - \frac{3}{2}\right)^2 + \frac{20}{4} - \frac{9}{4} \][/tex]
[tex]\[ \left(x - \frac{3}{2}\right)^2 + \frac{11}{4} \][/tex]
Thus, the expression is:
[tex]\[ \left(x - \frac{3}{2}\right)^2 + \frac{11}{4} \][/tex]
### 3. [tex]\(4x^2 + 8x + 12\)[/tex]
1. Factor out the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ 4(x^2 + 2x) + 12 \][/tex]
2. Now, complete the square inside the parentheses:
[tex]\[ \left(\frac{2}{2}\right)^2 = 1 \][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[ 4(x^2 + 2x + 1 - 1) + 12 \][/tex]
[tex]\[ 4((x + 1)^2 - 1) + 12 \][/tex]
[tex]\[ 4(x + 1)^2 - 4 + 12 \][/tex]
[tex]\[ 4(x + 1)^2 + 8 \][/tex]
Thus, the expression is:
[tex]\[ 4(x + 1)^2 + 8 \][/tex]
### 4. [tex]\(5x^2 - 3x + 13\)[/tex]
1. Factor out the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ 5(x^2 - \frac{3}{5}x) + 13 \][/tex]
2. Now, complete the square inside the parentheses:
[tex]\[ \left(\frac{-3/5}{2}\right)^2 = \left(\frac{-3}{10}\right)^2 = \frac{9}{100} \][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[ 5\left(x^2 - \frac{3}{5}x + \frac{9}{100} - \frac{9}{100}\right) + 13 \][/tex]
[tex]\[ 5\left(\left(x - \frac{3}{10}\right)^2 - \frac{9}{100}\right) + 13 \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 - \frac{45}{100} + 13 \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 - \frac{9}{20} + 13 \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 + \frac{260}{20} - \frac{9}{20} \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 + \frac{251}{20} \][/tex]
Thus, the expression is:
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 + \frac{251}{20} \][/tex]
In conclusion, the expressions are:
1. [tex]\( (x + 3)^2 - 19 \)[/tex]
2. [tex]\( \left(x - \frac{3}{2}\right)^2 + \frac{11}{4} \)[/tex]
3. [tex]\( 4(x + 1)^2 + 8 \)[/tex]
4. [tex]\( 5\left(x - \frac{3}{10}\right)^2 + \frac{251}{20} \)[/tex]
### 1. [tex]\(x^2 + 6x - 10\)[/tex]
1. Identify the coefficient of [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[ a = 1, \quad b = 6 \][/tex]
2. Half the coefficient of [tex]\(x\)[/tex], then square it.
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
3. Add and subtract this square inside the expression:
[tex]\[ x^2 + 6x + 9 - 9 - 10 \][/tex]
[tex]\[ (x + 3)^2 - 19 \][/tex]
Thus, the expression is:
[tex]\[ (x + 3)^2 - 19 \][/tex]
### 2. [tex]\(x^2 - 3x + 5\)[/tex]
1. Identify the coefficient of [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[ a = 1, \quad b = -3 \][/tex]
2. Half the coefficient of [tex]\(x\)[/tex], then square it.
[tex]\[ \left(\frac{-3}{2}\right)^2 = \left(\frac{-3}{2}\right)^2 = \frac{9}{4} \][/tex]
3. Add and subtract this square inside the expression:
[tex]\[ x^2 - 3x + \frac{9}{4} - \frac{9}{4} + 5 \][/tex]
[tex]\[ \left(x - \frac{3}{2}\right)^2 + 5 - \frac{9}{4} \][/tex]
[tex]\[ \left(x - \frac{3}{2}\right)^2 + \frac{20}{4} - \frac{9}{4} \][/tex]
[tex]\[ \left(x - \frac{3}{2}\right)^2 + \frac{11}{4} \][/tex]
Thus, the expression is:
[tex]\[ \left(x - \frac{3}{2}\right)^2 + \frac{11}{4} \][/tex]
### 3. [tex]\(4x^2 + 8x + 12\)[/tex]
1. Factor out the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ 4(x^2 + 2x) + 12 \][/tex]
2. Now, complete the square inside the parentheses:
[tex]\[ \left(\frac{2}{2}\right)^2 = 1 \][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[ 4(x^2 + 2x + 1 - 1) + 12 \][/tex]
[tex]\[ 4((x + 1)^2 - 1) + 12 \][/tex]
[tex]\[ 4(x + 1)^2 - 4 + 12 \][/tex]
[tex]\[ 4(x + 1)^2 + 8 \][/tex]
Thus, the expression is:
[tex]\[ 4(x + 1)^2 + 8 \][/tex]
### 4. [tex]\(5x^2 - 3x + 13\)[/tex]
1. Factor out the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ 5(x^2 - \frac{3}{5}x) + 13 \][/tex]
2. Now, complete the square inside the parentheses:
[tex]\[ \left(\frac{-3/5}{2}\right)^2 = \left(\frac{-3}{10}\right)^2 = \frac{9}{100} \][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[ 5\left(x^2 - \frac{3}{5}x + \frac{9}{100} - \frac{9}{100}\right) + 13 \][/tex]
[tex]\[ 5\left(\left(x - \frac{3}{10}\right)^2 - \frac{9}{100}\right) + 13 \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 - \frac{45}{100} + 13 \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 - \frac{9}{20} + 13 \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 + \frac{260}{20} - \frac{9}{20} \][/tex]
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 + \frac{251}{20} \][/tex]
Thus, the expression is:
[tex]\[ 5\left(x - \frac{3}{10}\right)^2 + \frac{251}{20} \][/tex]
In conclusion, the expressions are:
1. [tex]\( (x + 3)^2 - 19 \)[/tex]
2. [tex]\( \left(x - \frac{3}{2}\right)^2 + \frac{11}{4} \)[/tex]
3. [tex]\( 4(x + 1)^2 + 8 \)[/tex]
4. [tex]\( 5\left(x - \frac{3}{10}\right)^2 + \frac{251}{20} \)[/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
