To write a quadratic equation for the parabola that passes through the point (-2, 12) and has roots at [tex]\( x = -5 \)[/tex] and [tex]\( x = -3 \)[/tex], follow these steps:
### Step 1: General Form of the Quadratic Equation
For a quadratic equation with roots [tex]\( x = r_1 \)[/tex] and [tex]\( x = r_2 \)[/tex], the general form is:
[tex]\[ y = a(x - r_1)(x - r_2) \][/tex]
Here, the given roots are [tex]\( r_1 = -5 \)[/tex] and [tex]\( r_2 = -3 \)[/tex]. Substituting these into the general form:
[tex]\[ y = a(x + 5)(x + 3) \][/tex]
### Step 2: Use the Given Point to Find the Value of [tex]\(a\)[/tex]
You are given that the parabola passes through the point [tex]\((-2, 12)\)[/tex]. This means that when [tex]\( x = -2 \)[/tex], [tex]\( y = 12 \)[/tex].
Substitute this point into the quadratic equation to find [tex]\( a \)[/tex]:
[tex]\[ 12 = a(-2 + 5)(-2 + 3) \][/tex]
Simplify inside the parentheses:
[tex]\[ 12 = a(3)(1) \][/tex]
[tex]\[ 12 = 3a \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{12}{3} \][/tex]
[tex]\[ a = 4 \][/tex]
### Step 3: Write the Final Quadratic Equation
Now that you have found [tex]\( a = 4 \)[/tex], substitute it back into the general form of the quadratic equation:
[tex]\[ y = 4(x + 5)(x + 3) \][/tex]
### Step 4: Expand the Quadratic Equation (Optional)
You can expand this equation to get it in the standard form [tex]\( y = ax^2 + bx + c \)[/tex] if desired. First, expand the factors:
[tex]\[ y = 4[(x + 5)(x + 3)] \][/tex]
[tex]\[ y = 4[x^2 + 3x + 5x + 15] \][/tex]
[tex]\[ y = 4[x^2 + 8x + 15] \][/tex]
[tex]\[ y = 4x^2 + 32x + 60 \][/tex]
Thus, the quadratic equation can also be written in standard form as:
[tex]\[ y = 4x^2 + 32x + 60 \][/tex]
So, the quadratic equation for the parabola that passes through the point [tex]\((-2, 12)\)[/tex] and has roots at [tex]\(-5\)[/tex] and [tex]\(-3\)[/tex] is:
[tex]\[ y = 4(x + 5)(x + 3) \][/tex]
or in expanded form:
[tex]\[ y = 4x^2 + 32x + 60 \][/tex]
