To find the quadratic equation based on the given conditions, let's name the roots of the equation as [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex].
### Step 1: Using the Sum of the Roots
The sum of the roots [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] is given as [tex]\( \alpha + \beta = 7 \)[/tex].
### Step 2: Using the Sum of the Squares of the Roots
We are also given that the sum of the squares of the roots is [tex]\( 25 \)[/tex]. Mathematically, we can express this as:
[tex]\[ \alpha^2 + \beta^2 = 25 \][/tex]
### Step 3: Relate the Sum and Product of the Roots
Using the identities for the roots of a quadratic equation:
[tex]\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \][/tex]
Substitute the values we know:
[tex]\[ 25 = 7^2 - 2\alpha\beta \][/tex]
[tex]\[ 25 = 49 - 2\alpha\beta \][/tex]
### Step 4: Solve for Product of the Roots
Rearrange the equation to find [tex]\( \alpha\beta \)[/tex]:
[tex]\[ 25 = 49 - 2\alpha\beta \][/tex]
[tex]\[ 2\alpha\beta = 49 - 25 \][/tex]
[tex]\[ 2\alpha\beta = 24 \][/tex]
[tex]\[ \alpha\beta = 12 \][/tex]
### Step 5: Form the Quadratic Equation
A quadratic equation with roots [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] can be written as:
[tex]\[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \][/tex]
Substitute the values obtained for [tex]\( \alpha + \beta \)[/tex] and [tex]\( \alpha\beta \)[/tex]:
[tex]\[ x^2 - 7x + 12 = 0 \][/tex]
Thus, the quadratic equation is:
[tex]\[ \boxed{x^2 - 7x + 12 = 0} \][/tex]
