5.) How many, and what type of solutions does 2x^2 + 11x + 15 = 0 have?
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Answer:
Step-by-step explanation:
We want to write the equation of a quadratic whose graph passes through (-3, 2), (-1, 0), and (1, 6).
Remember that the standard quadratic function is given by:
Since it passes through the point (-3, 2). This means that when ,
. Hence:
Simplify:
Perform the same computations for the coordinates (-1, 0) and (1, 6). Therefore:
And for (1, 6):
So, we have a triple system of equations:
We can solve this using elimination.
Notice that the b term in Equation 2 and 3 are opposites. Hence, let's add them together. This yields:
Compute:
Let's divide both sides by 2:
Now, let's eliminate b again but we will use Equation 1 and 2.
Notice that if we multiply Equation 2 by -3, then the b terms will be opposites. So:
Multiply:
Add this to Equation 1:
Compute:
Again, we can divide both sides by 2:
So, we know have two equations with only two variables:
We can solve for a using elimination since the c term are opposites of each other. Add the two equations together:
Compute:
Solve for a:
So, the value of a is 1.
Using either of the two equations, we can now find c. Let's use the first one. Hence:
Substitute 1 for a and solve for c:
So, the value of c is 2.
Finally, using any of the three original equations, solve for b:
We can use Equation 3. Hence:
Substitute in known values and solve for b:
Therefore, a=1, b=3, and c=2.
Hence, our quadratic function is: