Write the equation of a quadratic function with the given solutions. Use standard form \( f(x)=a x^{2}+b x+c \) \( x=2 \) and \( x=1 \) Show your work here Hint: To add an exponent \( \left(x^{y}\right) \), type "exponent" or press " \( \wedge \) "

To find the quadratic function with solutions \( x = 2 \) and \( x = 1 \), we start with the fact that if \( x = p \) and \( x = q \) are solutions, then the quadratic can be expressed as: \[ f(x) = a(x - p)(x - q) \] Plugging in our solutions, we have: \[ f(x) = a(x - 2)(x - 1) \] Expanding this, we get: \[ f(x) = a[(x - 2)(x - 1)] = a(x^2 - 3x + 2) \] Now, to express it in standard form \( f(x) = ax^2 + bx + c \), we simply distribute \( a \): \[ f(x) = ax^2 - 3ax + 2a \] For \( a = 1 \) (to keep it simple), the quadratic function becomes: \[ f(x) = x^2 - 3x + 2 \] Thus, \( f(x) = x^2 - 3x + 2 \) is the quadratic function in standard form with the given solutions. Check your work: The solutions can be found by setting \( f(x) = 0 \) and factoring: \[ (x - 2)(x - 1) = 0 \] This hence confirms the solutions are indeed \( x = 2 \) and \( x = 1 \).