Characteristics of Quadratic Equations Which of the following statements are true about the horizontal intercepts of \( a \) Quadratic Equation in Standard Form? Check all that apply. A QUADRATIC EQUATION in STANDARD FORM is an equation of the form \( f(x)=a x^{2}+x+c \) A QUADRATIC EQUATION in STANDARD FORM is an equation of the form \( a x^{2}+b x+c=0 \) 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 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 ( \( \left.x_{2}, 0\right) \) There are three possible cases for the number of solutions to a quadratic equation in standard form 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 \( \left(x_{1}, 0\right) \)

1. **Statement 1:** The statement claims that a quadratic equation in standard form is given by \[ f(x)=ax^2+ x+ c. \] This form omits the coefficient (typically denoted by \( b \)) in front of \( x \). The standard form for a quadratic function or equation is generally \[ f(x)=ax^2+bx+c \quad \text{or} \quad ax^2+bx+c=0. \] **Conclusion:** This statement is **false**. 2. **Statement 2:** The statement asserts that a quadratic equation in standard form is \[ ax^2+bx+c=0. \] This is indeed the standard form for quadratic equations where \( a \neq 0 \). **Conclusion:** This statement is **true**. 3. **Statement 3:** The statement says, "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." For a quadratic function, if the discriminant \( b^2-4ac<0 \), the equation has complex (non-real) solutions, and consequently, the graph (a parabola) does not intersect the \( x \)-axis. **Conclusion:** This statement is **true**. 4. **Statement 4:** The statement indicates that if \[ ax^2+bx+c=0 \] has real number solutions \( x_1 \) and \( x_2 \), then the \( x \)-intercepts of \[ f(x)=ax^2+bx+c \] are given by \(\left(x_1,0\right)\) and \(\left(x_2,0\right)\). This is the correct interpretation for the points where the graph of the function crosses the \( x \)-axis. **Conclusion:** This statement is **true**. 5. **Statement 5:** The statement claims, "There are three possible cases for the number of solutions to a quadratic equation in standard form." The three cases typically refer to: - Two distinct real solutions (when \( b^2-4ac>0 \)). - One repeated real solution (when \( b^2-4ac=0 \)). - Two complex conjugate solutions (when \( b^2-4ac<0 \)). This encompasses all possible cases for the nature of the solutions of a quadratic equation. **Conclusion:** This statement is **true**. 6. **Statement 6:** The statement says, "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 \(\left(x_1, 0\right)\)." When a parabola touches the \( x \)-axis, the discriminant is zero, which gives a repeated real solution (i.e., a double root). Although it is technically true that all real numbers are part of the complex number system, the usual interpretation here is that the solution is a real number, not a non-real complex number. Additionally, saying that the solutions lie in the complex number system in this context is misleading because it implies non-real complex solutions, which is not the case when the parabola touches the \( x \)-axis. **Conclusion:** This statement is **false**. **Final Answer:** Statements 2, 3, 4, and 5 are true.