Respuesta :
Given:
[tex]x^2+2x-12=3[/tex]To solve the questions, evaluate each statement.
(a) The standard form of the equation.
The standard form of a quadratic equation is ax² + bx + c = 0. So, to find the standard form, subtract 3 from both sides of the equation.
[tex]\begin{gathered} x^2+2x-12-3=3-3 \\ x^2+2x-15=0 \end{gathered}[/tex]So, the statement is correct.
(b) The factored form.
The factored form of a quadratic equation is (x-a)(x-b) = 0, where a and b are the zeros of the equations.
To find the zeros, use the quadratic formula.
For ax² + bx + c = 0, the zeros are:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}^{}[/tex]So, substituting the values:
[tex]\begin{gathered} x=\frac{-2\pm\sqrt[]{2^2-4\cdot1\cdot(-15)}}{2\cdot1}^{} \\ x=\frac{-2\pm\sqrt[]{4+60}}{2}^{}=\frac{-2\pm\sqrt[]{64}}{2} \\ x=\frac{-2\pm8}{2} \\ x_1=\frac{-2-8}{2}=-\frac{10}{2}=-5 \\ x_2=\frac{-2+8}{2}=\frac{6}{2}=3 \end{gathered}[/tex]The zeros are -5 and 3. So, the factored form is (x-3)(x+5) = 0
So, the statement is correct.
(c) Zero-Product Property.
Since (x-3)(x+5) = 0, then:
(x - 3) = 0
or
(x + 5) = 0
So, the statement is correct.
(d) Solutions of the equation.
The solutions of the equation are the zeros: -5 and 3 (shown in part B).
Thus, this statement is false. The correct would be the solution is -5 and 3.
(e) x-coordinate of the vertex.
The x-coordinate of the vertex is:
[tex]\begin{gathered} x_v=\frac{-b}{2a} \\ x_v=\frac{-2}{2\cdot1} \\ x_v=\frac{-2}{2} \\ x_v=-1 \end{gathered}[/tex]So, the statement is correct.
Answer: Statement D is wrong. The correct statement is: the solution is -5 and 3.
