Write the equation of a quadratic function with the given solutions. Use standard form \( f(x)=a x^{2}+b x+c \). \( x=-3 \) and \( x=-10 \) Show your work here

To write the quadratic function with the given solutions \( x = -3 \) and \( x = -10 \), we can use the fact that if the solutions (roots) of the quadratic are \( r_1 \) and \( r_2 \), the quadratic can be expressed in factored form as: \[ f(x) = a(x - r_1)(x - r_2) \] Substituting the roots into the equation: \[ f(x) = a(x + 3)(x + 10) \] Next, we need to expand this to get it into standard form. First, we can expand the factors: \[ f(x) = a[(x + 3)(x + 10)] \] \[ = a(x^2 + 10x + 3x + 30) \] \[ = a(x^2 + 13x + 30) \] Now, multiplying through by \( a \): \[ f(x) = ax^2 + 13ax + 30a \] For the purposes of obtaining the standard form, we can choose \( a = 1 \) (but any non-zero constant would work, resulting in a family of functions), leading to: \[ f(x) = x^2 + 13x + 30 \] Thus, the quadratic function in standard form is: \[ f(x) = x^2 + 13x + 30 \]