The roots of the equation ax² + bx + c = 0 are -2 and -3/2. Find the values of a, b, and c.

Understand the Problem

The question is asking to find the values of a, b, and c for a quadratic equation given its roots. The roots are provided as -2 and -3/2, and the equation is in standard form ax² + bx + c = 0.

Answer

The values are $a = 1$, $b = \frac{7}{2}$, and $c = 3$.

Steps to Solve

Identify the roots The given roots of the equation are $r_1 = -2$ and $r_2 = -\frac{3}{2}$.
Use the factored form of a quadratic equation The standard form of a quadratic equation can be expressed as $$y = a(x - r_1)(x - r_2)$$ Substituting the roots into the equation gives: $$y = a(x + 2)\left(x + \frac{3}{2}\right)$$
Multiply the factors First, multiply $(x + 2)$ and $(x + \frac{3}{2})$: [ (x + 2)\left(x + \frac{3}{2}\right) = x^2 + \frac{3}{2}x + 2x + 3 = x^2 + \frac{7}{2}x + 3 ]
Include the leading coefficient Now, we multiply by $a$: $$y = a\left(x^2 + \frac{7}{2}x + 3\right)$$
Choose a value for a For simplicity, let's set $a = 1$. The equation simplifies to: $$y = x^2 + \frac{7}{2}x + 3$$
Identify coefficients a, b, and c From the equation $y = x^2 + \frac{7}{2}x + 3$, we can identify:

$a = 1$
$b = \frac{7}{2}$
$c = 3$

The values are

$a = 1$
$b = \frac{7}{2}$
$c = 3$.

More Information

Setting the leading coefficient $a$ to 1 is common practice, but you can choose any non-zero value for $a$. The quadratic equation will provide the same roots.

Tips

Forgetting to multiply the factors correctly.
Not simplifying the factored form.
Misidentifying the values of $a$, $b$, and $c$ from the expanded equation.

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