\begin{tabular}{l} Characteristics of Quadratic Equations \\ \hline \( \begin{array}{l}\text { Which of the following statements are true about the horizontal intercepts of a } \\ \text { Quadratic Equation in Standard Form? Check all that apply. }\end{array} \) \\ \hline If a parabola does not cross the \( x \)-axis, then its solutions lie in the complex number \\ system and we say that it has no real x-intercepts \\ If a parabola touches, but does not cross the \( x \)-axis, then its solutions lie in the \\ complex number system and the x-intercept is ( \( x_{1}, 0 \) ) \\ A QUADRATIC EQUATION in STANDARD FORM is an equation of the form \\ \( a x^{2}+b x+c=0 \) \\ A QUADRATIC EQUATION in STANDARD FORM is an equation of the form \\ \( f(x)=a x^{2}+x+c \) \\ There are three possible cases for the number of solutions to a quadratic equation \\ in standard form \\ If the quadratic equation \( a x^{2}+b x+c=0 \) has real number solutions \( x_{1} \) and \( x_{2} \) \\ then the x-intercepts of \( f(x)=a x^{2}+b x+c \) are \( \left(x_{1}, 0\right) \) and ( \( \left.x_{2}, 0\right) \)\end{tabular}

Below is an analysis of each statement: 1. “If a parabola does not cross the x‑axis, then its solutions lie in the complex number system and we say that it has no real x‑intercepts.”  • This is true. When the parabola does not intersect the x‑axis the discriminant is negative, and the quadratic’s solutions are non‐real complex numbers (even though all real numbers are technically complex, here the contrast is between “real” and “non‑real” solutions). 2. “If a parabola touches, but does not cross the x‑axis, then its solutions lie in the complex number system and the x‑intercept is (x₁, 0).”  • This is false. When the parabola touches the x‑axis it has exactly one unique (repeated) real solution. Although every real number is a complex number when viewed abstractly, in this context “complex number system” is used to distinguish non‑real solutions from real ones. The correct interpretation is that the equation has one repeated real solution, not non‑real ones. 3. “A QUADRATIC EQUATION in STANDARD FORM is an equation of the form ax² + bx + c = 0.”  • This is true. 4. “A QUADRATIC EQUATION in STANDARD FORM is an equation of the form f(x) = ax² + x + c.”  • This is false. The standard form must include the bx term, not x (which would imply b = 1). The correct general form is ax²+bx+c (equal to 0 if written as an equation to solve). 5. “There are three possible cases for the number of solutions to a quadratic equation in standard form.”  • This is true. The quadratic can have: two distinct real solutions (when the discriminant is positive), one real solution (when the discriminant is zero), or two non‑real complex solutions (when the discriminant is negative). 6. “If the quadratic equation ax² + bx + c = 0 has real number solutions x₁ and x₂ then the x‑intercepts of f(x) = ax² + bx + c are (x₁, 0) and (x₂, 0).”  • This is true. Therefore, the correct responses (the true statements regarding horizontal intercepts) are statements 1, 3, 5, and 6.