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