When solving x^2 – 3x – 18 = 0 using the quadratic formula, what is under the radical?
The correct answer and explanation is:
Correct Answer: 81
To solve the quadratic equation x2−3x−18=0x^2 – 3x – 18 = 0 using the quadratic formula, we first recall the formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
In this equation, the coefficients are:
- a=1a = 1
- b=−3b = -3
- c=−18c = -18
The expression under the radical is called the discriminant, and it is represented by: b2−4acb^2 – 4ac
Substitute the values of aa, bb, and cc: (−3)2−4(1)(−18)(-3)^2 – 4(1)(-18)
First, square −3-3: 9−4(1)(−18)9 – 4(1)(-18)
Now, calculate 4×1×−18=−724 \times 1 \times -18 = -72. So: 9−(−72)=9+72=819 – (-72) = 9 + 72 = 81
Thus, the value under the radical (the discriminant) is 81.
The discriminant tells us about the nature of the solutions:
- If the discriminant is positive and a perfect square (like 81), the equation has two real and rational solutions.
- If the discriminant is positive but not a perfect square, the equation has two real but irrational solutions.
- If it is zero, the equation has one real and repeated root.
- If it is negative, the solutions are complex (non-real).
Since 81 is a positive perfect square, we expect two real and rational roots.
To summarize, the value under the radical in the quadratic formula for the equation x2−3x−18=0x^2 – 3x – 18 = 0 is 81, and it helps determine the type of roots the equation has.