Answer :
To solve the quadratic equation [tex]\( x^2 - 9x + 8 = 0 \)[/tex] using the form [tex]\( x^2 + (a + b)x + ab = 0 \)[/tex], follow these steps:
1. Identify the coefficients in the given quadratic equation:
The given quadratic equation is:
[tex]\[ x^2 - 9x + 8 = 0 \][/tex]
By comparing it with the standard form [tex]\( x^2 + (a + b)x + ab = 0 \)[/tex], we can identify that:
[tex]\[ a + b = 9 \][/tex]
[tex]\[ ab = 8 \][/tex]
2. Formulate the system of equations:
We now have the following system of equations:
[tex]\[ a + b = 9 \][/tex]
[tex]\[ ab = 8 \][/tex]
3. Solve the system of equations:
To solve this system, we need to find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy both equations simultaneously.
4. Determine the pair values:
One pair of values that satisfy these equations is [tex]\( a = 1 \)[/tex] and [tex]\( b = 8 \)[/tex]:
[tex]\[ a + b = 1 + 8 = 9 \][/tex]
[tex]\[ ab = 1 \times 8 = 8 \][/tex]
Another pair of values that satisfy the equations is [tex]\( a = 8 \)[/tex] and [tex]\( b = 1 \)[/tex]:
[tex]\[ a + b = 8 + 1 = 9 \][/tex]
[tex]\[ ab = 8 \times 1 = 8 \][/tex]
Therefore, the solutions to the system of equations are:
[tex]\[ (a, b) = (1, 8) \quad \text{or} \quad (a, b) = (8, 1) \][/tex]
1. Identify the coefficients in the given quadratic equation:
The given quadratic equation is:
[tex]\[ x^2 - 9x + 8 = 0 \][/tex]
By comparing it with the standard form [tex]\( x^2 + (a + b)x + ab = 0 \)[/tex], we can identify that:
[tex]\[ a + b = 9 \][/tex]
[tex]\[ ab = 8 \][/tex]
2. Formulate the system of equations:
We now have the following system of equations:
[tex]\[ a + b = 9 \][/tex]
[tex]\[ ab = 8 \][/tex]
3. Solve the system of equations:
To solve this system, we need to find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy both equations simultaneously.
4. Determine the pair values:
One pair of values that satisfy these equations is [tex]\( a = 1 \)[/tex] and [tex]\( b = 8 \)[/tex]:
[tex]\[ a + b = 1 + 8 = 9 \][/tex]
[tex]\[ ab = 1 \times 8 = 8 \][/tex]
Another pair of values that satisfy the equations is [tex]\( a = 8 \)[/tex] and [tex]\( b = 1 \)[/tex]:
[tex]\[ a + b = 8 + 1 = 9 \][/tex]
[tex]\[ ab = 8 \times 1 = 8 \][/tex]
Therefore, the solutions to the system of equations are:
[tex]\[ (a, b) = (1, 8) \quad \text{or} \quad (a, b) = (8, 1) \][/tex]
