ax^(2)+5x+2=0 In the given equation, a is a constant. If the equation has the solutions x=-2 and x=-(1)/(2), what is the value of a ?

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ax^(2)+5x+2=0 In the given equation,  a is a constant. If the equation has the solutions  x=-2 and  x=-(1)/(2), what is the value of  a ?

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Use Factored Form: Let's use the fact that if x = -2 and x = -(1/2) are solutions to the equation ax^2 + 5x + 2 = 0, then the equation can be factored as a(x + 2)(x + 1/2) = 0.Expand and Compare: We can expand the factored form to get the standard form of the quadratic equation. Let's do that: a(x + 2)(x + 1/2) = ax(x + 1/2) + 2a(x + 1/2) = ax^2 + (1/2)ax + 2ax + a = ax^2 + (5/2)a + 2ax + a.Set Up Equations: Now, we compare the coefficients from the expanded form with the original equation ax^2 + 5x + 2 = 0. We have: Coefficient of x^2: a (from both equations) Coefficient of x: (5/2)a + 2a (from the expanded form) and 5 (from the original equation) Constant term: a (from the expanded form) and 2 (from the original equation)Solve for a: Let's set up the equations for the coefficients of x and the constant term: (5/2)a + 2a = 5 a = 2Check Constant Term: Now, we solve for a. Combine like terms in the coefficient equation: (5/2)a + 2a = (5/2 + 4/2)a = (9/2)a = 5Multiply both sides by 2/9 to solve for a: a = (5 * 2/9) a = 10/9Now, let's check if the constant term also matches with a = 10/9: a = 10/9 The constant term in the expanded form is also a, which should be equal to 2. So, 10/9 = 2, which is not true. This means there is a mistake in our previous steps.

$a x^{2}+5 x+2=0$ $\newline$ In the given equation, $a$ is a constant. If the equation has the solutions $x=-2$ and $x=-\frac{1}{2}$ , what is the value of $a$ ?

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ax2+5x+2=0 a x^{2}+5 x+2=0 ax2+5x+2=0\newlineIn the given equation, a a a is a constant. If the equation has the solutions x=−2 x=-2 x=−2 and x=−12 x=-\frac{1}{2} x=−21​, what is the value of a a a ?

$a x^{2}+5 x+2=0$ $\newline$ In the given equation, $a$ is a constant. If the equation has the solutions $x=-2$ and $x=-\frac{1}{2}$ , what is the value of $a$ ?