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]
