ax2+bx+c=0 let x=1a+b+c=0a+b=-c x 4 4a+4b=-4c(-)4a+2b=-c2b=-3cc=-2b/3

Understanding the Problem
To find the values of the unknown constants \( a \), \( b \), and \( c \) in the quadratic equation \( ax^2 + bx + c = 0 \) when the solutions are \( x = 1 \) and \( x = 2 \), we can apply Vieta's formulas.
Using Vieta's Formulas
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \) with roots \( r_1 \) and \( r_2 \):
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
In this case, our roots are \( x = 1 \) and \( x = 2 \).
Calculating the Sum and Product
- Sum of roots:
\( r_1 + r_2 = 1 + 2 = 3 \)
- Product of roots:
\( r_1 \cdot r_2 = 1 \cdot 2 = 2 \)
Setting Up the Equations
Using Vieta's formulas, we can set up the following equations:
- From the sum of roots:
\( 3 = -\frac{b}{a} \)
This implies \( b = -3a \)
- From the product of roots:
\( 2 = \frac{c}{a} \)
This gives \( c = 2a \)
Choosing a Value for a
To find specific values for \( b \) and \( c \), we can choose a value for \( a \). A common choice is \( a = 1 \):
- If \( a = 1 \):
- \( b = -3 \times 1 = -3 \)
- \( c = 2 \times 1 = 2 \)
Thus, the quadratic equation can be expressed as:
Final Equation
The equation is:
\[ x^2 - 3x + 2 = 0 \]
This confirms that the roots \( x = 1 \) and \( x = 2 \) satisfy the equation.