Answer :
To solve the given quadratic equations by extracting the square roots, let's consider each equation individually:
1. [tex]\( G: x^2 - 16x + 64 = 144 \)[/tex]
First, we simplify the equation:
[tex]\[ x^2 - 16x + 64 - 144 = 0 \][/tex]
[tex]\[ x^2 - 16x - 80 = 0 \][/tex]
To solve this quadratic equation, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 1, \, b = -16, \, c = -80 \][/tex]
[tex]\[ x = \frac{16 \pm \sqrt{256 + 320}}{2} \][/tex]
[tex]\[ x = \frac{16 \pm \sqrt{576}}{2} \][/tex]
[tex]\[ x = \frac{16 \pm 24}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{16 + 24}{2} = 20 \][/tex]
[tex]\[ x = \frac{16 - 24}{2} = -4 \][/tex]
So, the solutions are:
[tex]\[ x = 20 \text{ and } x = -4 \][/tex]
2. [tex]\( N: x^2 - x + \frac{1}{4} = \frac{4}{9} \)[/tex]
First, we simplify the equation:
[tex]\[ x^2 - x + \frac{1}{4} - \frac{4}{9} = 0 \][/tex]
To combine the fractions, we find a common denominator:
[tex]\[ x^2 - x + \left( \frac{9}{36} - \frac{16}{36} \right) = 0 \][/tex]
[tex]\[ x^2 - x - \frac{7}{36} = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 1, \, b = -1, \, c = -\frac{7}{36} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{1 + \frac{28}{36}}}{2} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{\frac{64}{36}}}{2} \][/tex]
[tex]\[ x = \frac{1 \pm \frac{8}{6}}{2} \][/tex]
[tex]\[ x = \frac{1 \pm \frac{4}{3}}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{1 + \frac{4}{3}}{2} = \frac{7}{6} \][/tex]
[tex]\[ x = \frac{1 - \frac{4}{3}}{2} = -\frac{1}{6} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{7}{6} \text{ and } x = -\frac{1}{6} \][/tex]
3. [tex]\( A: 4x^2 - 20x + 25 = 49 \)[/tex]
First, we simplify the equation:
[tex]\[ 4x^2 - 20x + 25 - 49 = 0 \][/tex]
[tex]\[ 4x^2 - 20x - 24 = 0 \][/tex]
Again, using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 4, \, b = -20, \, c = -24 \][/tex]
[tex]\[ x = \frac{20 \pm \sqrt{400 + 384}}{8} \][/tex]
[tex]\[ x = \frac{20 \pm \sqrt{784}}{8} \][/tex]
[tex]\[ x = \frac{20 \pm 28}{8} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{20 + 28}{8} = 6 \][/tex]
[tex]\[ x = \frac{20 - 28}{8} = -1 \][/tex]
So, the solutions are:
[tex]\[ x = 6 \text{ and } x = -1 \][/tex]
4. [tex]\( c: 9x^2 = 25 \)[/tex]
First, we simplify the equation:
[tex]\[ 9x^2 - 25 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we know that:
[tex]\[ x^2 = \frac{25}{9} \][/tex]
[tex]\[ x = \pm \frac{5}{3} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{5}{3} \text{ and } x = -\frac{5}{3} \][/tex]
5. [tex]\( H: x^2 + 6 = 10 \)[/tex]
First, we simplify the equation:
[tex]\[ x^2 + 6 - 10 = 0 \][/tex]
[tex]\[ x^2 - 4 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x^2 = 4 \][/tex]
[tex]\[ x = \pm 2 \][/tex]
So, the solutions are:
[tex]\[ x = 2 \text{ and } x = -2 \][/tex]
6. [tex]\( E: 9x^2 - 12x + 4 = 4 \)[/tex]
First, we simplify the equation:
[tex]\[ 9x^2 - 12x + 4 - 4 = 0 \][/tex]
[tex]\[ 9x^2 - 12x = 0 \][/tex]
Solving the quadratic equation, we factor out common terms:
[tex]\[ x(9x - 12) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x = 0 \][/tex]
[tex]\[ 9x - 12 = 0 \Rightarrow x = \frac{12}{9} = \frac{4}{3} \][/tex]
So, the solutions are:
[tex]\[ x = 0 \text{ and } x = \frac{4}{3} \][/tex]
Matching solutions with the decoder:
- [tex]\( G: x = 20, -4 \)[/tex] => matches [tex]\( 20, -4 \)[/tex]
- [tex]\( N: x = \frac{7}{6}, -\frac{1}{6} \)[/tex] => matches [tex]\( \frac{7}{6}, -\frac{1}{6} \)[/tex]
- [tex]\( A: x = 6, -1 \)[/tex] => matches [tex]\( 6, -1 \)[/tex]
- [tex]\( c: x = \pm \frac{5}{3} \)[/tex] => matches [tex]\( \pm \frac{5}{3} \)[/tex]
- [tex]\( H: x = \pm 2 \)[/tex] => matches [tex]\( \pm 2 \)[/tex]
- [tex]\( E: x = 0, \frac{4}{3} \)[/tex] => matches [tex]\( 0, \frac{4}{3} \)[/tex]
So, the correct letters corresponding to solutions are:
- [tex]\( G \)[/tex]: [tex]\( 20, -4 \)[/tex]
- [tex]\( N \)[/tex]: [tex]\( \frac{7}{6}, -\frac{1}{6} \)[/tex]
- [tex]\( A \)[/tex]: [tex]\( 6, -1 \)[/tex]
- [tex]\( c \)[/tex]: [tex]\( \pm \frac{5}{3} \)[/tex]
- [tex]\( H \)[/tex]: [tex]\( \pm 2 \)[/tex]
- [tex]\( E \)[/tex]: [tex]\( 0, \frac{4}{3} \)[/tex]
1. [tex]\( G: x^2 - 16x + 64 = 144 \)[/tex]
First, we simplify the equation:
[tex]\[ x^2 - 16x + 64 - 144 = 0 \][/tex]
[tex]\[ x^2 - 16x - 80 = 0 \][/tex]
To solve this quadratic equation, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 1, \, b = -16, \, c = -80 \][/tex]
[tex]\[ x = \frac{16 \pm \sqrt{256 + 320}}{2} \][/tex]
[tex]\[ x = \frac{16 \pm \sqrt{576}}{2} \][/tex]
[tex]\[ x = \frac{16 \pm 24}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{16 + 24}{2} = 20 \][/tex]
[tex]\[ x = \frac{16 - 24}{2} = -4 \][/tex]
So, the solutions are:
[tex]\[ x = 20 \text{ and } x = -4 \][/tex]
2. [tex]\( N: x^2 - x + \frac{1}{4} = \frac{4}{9} \)[/tex]
First, we simplify the equation:
[tex]\[ x^2 - x + \frac{1}{4} - \frac{4}{9} = 0 \][/tex]
To combine the fractions, we find a common denominator:
[tex]\[ x^2 - x + \left( \frac{9}{36} - \frac{16}{36} \right) = 0 \][/tex]
[tex]\[ x^2 - x - \frac{7}{36} = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 1, \, b = -1, \, c = -\frac{7}{36} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{1 + \frac{28}{36}}}{2} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{\frac{64}{36}}}{2} \][/tex]
[tex]\[ x = \frac{1 \pm \frac{8}{6}}{2} \][/tex]
[tex]\[ x = \frac{1 \pm \frac{4}{3}}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{1 + \frac{4}{3}}{2} = \frac{7}{6} \][/tex]
[tex]\[ x = \frac{1 - \frac{4}{3}}{2} = -\frac{1}{6} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{7}{6} \text{ and } x = -\frac{1}{6} \][/tex]
3. [tex]\( A: 4x^2 - 20x + 25 = 49 \)[/tex]
First, we simplify the equation:
[tex]\[ 4x^2 - 20x + 25 - 49 = 0 \][/tex]
[tex]\[ 4x^2 - 20x - 24 = 0 \][/tex]
Again, using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = 4, \, b = -20, \, c = -24 \][/tex]
[tex]\[ x = \frac{20 \pm \sqrt{400 + 384}}{8} \][/tex]
[tex]\[ x = \frac{20 \pm \sqrt{784}}{8} \][/tex]
[tex]\[ x = \frac{20 \pm 28}{8} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{20 + 28}{8} = 6 \][/tex]
[tex]\[ x = \frac{20 - 28}{8} = -1 \][/tex]
So, the solutions are:
[tex]\[ x = 6 \text{ and } x = -1 \][/tex]
4. [tex]\( c: 9x^2 = 25 \)[/tex]
First, we simplify the equation:
[tex]\[ 9x^2 - 25 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we know that:
[tex]\[ x^2 = \frac{25}{9} \][/tex]
[tex]\[ x = \pm \frac{5}{3} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{5}{3} \text{ and } x = -\frac{5}{3} \][/tex]
5. [tex]\( H: x^2 + 6 = 10 \)[/tex]
First, we simplify the equation:
[tex]\[ x^2 + 6 - 10 = 0 \][/tex]
[tex]\[ x^2 - 4 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x^2 = 4 \][/tex]
[tex]\[ x = \pm 2 \][/tex]
So, the solutions are:
[tex]\[ x = 2 \text{ and } x = -2 \][/tex]
6. [tex]\( E: 9x^2 - 12x + 4 = 4 \)[/tex]
First, we simplify the equation:
[tex]\[ 9x^2 - 12x + 4 - 4 = 0 \][/tex]
[tex]\[ 9x^2 - 12x = 0 \][/tex]
Solving the quadratic equation, we factor out common terms:
[tex]\[ x(9x - 12) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x = 0 \][/tex]
[tex]\[ 9x - 12 = 0 \Rightarrow x = \frac{12}{9} = \frac{4}{3} \][/tex]
So, the solutions are:
[tex]\[ x = 0 \text{ and } x = \frac{4}{3} \][/tex]
Matching solutions with the decoder:
- [tex]\( G: x = 20, -4 \)[/tex] => matches [tex]\( 20, -4 \)[/tex]
- [tex]\( N: x = \frac{7}{6}, -\frac{1}{6} \)[/tex] => matches [tex]\( \frac{7}{6}, -\frac{1}{6} \)[/tex]
- [tex]\( A: x = 6, -1 \)[/tex] => matches [tex]\( 6, -1 \)[/tex]
- [tex]\( c: x = \pm \frac{5}{3} \)[/tex] => matches [tex]\( \pm \frac{5}{3} \)[/tex]
- [tex]\( H: x = \pm 2 \)[/tex] => matches [tex]\( \pm 2 \)[/tex]
- [tex]\( E: x = 0, \frac{4}{3} \)[/tex] => matches [tex]\( 0, \frac{4}{3} \)[/tex]
So, the correct letters corresponding to solutions are:
- [tex]\( G \)[/tex]: [tex]\( 20, -4 \)[/tex]
- [tex]\( N \)[/tex]: [tex]\( \frac{7}{6}, -\frac{1}{6} \)[/tex]
- [tex]\( A \)[/tex]: [tex]\( 6, -1 \)[/tex]
- [tex]\( c \)[/tex]: [tex]\( \pm \frac{5}{3} \)[/tex]
- [tex]\( H \)[/tex]: [tex]\( \pm 2 \)[/tex]
- [tex]\( E \)[/tex]: [tex]\( 0, \frac{4}{3} \)[/tex]
