Filling the Gap: Determine the Unknown Coefficient to Make -1 a Solution

Question

Look at the following equation:

7x2+□x+2=0 7x^2+\square x+2=0

Fill in the blank so that one of the solutions to the equation is -1.

Video Solution

Solution Steps

00:10 Let's complete it so one of the answers will be negative one.
00:14 The unknown here is the coefficient, which we'll call B.
00:21 First, let's look at the coefficients together.
00:27 We'll use the root formula to discover possible solutions.
00:35 By substituting values, we'll solve and find these solutions.
00:52 We'll compare two options: subtraction and addition.
01:07 Suppose the first option equals negative one.
01:15 We multiply by the denominator to clear the fraction.
01:23 Next, let's put the root by itself.
01:28 Since the root is greater than zero, this is where B can be defined.
01:49 Now, let's square both sides to get rid of the root.
01:58 We'll expand using the shortened multiplication formulas.
02:08 Time to gather the terms together.
02:18 Then, let's get B all by itself.
02:25 Uh-oh, this solution isn't valid as it's not in B's defined range.
02:29 Let's try the same steps for the second option now.
02:39 Again, multiply by the denominator to remove the fraction.
02:50 Separate the root by itself once more.
02:55 Remember, the root is greater than zero, defining B's space.
03:05 Square both sides again to lose the root.
03:13 Expand using our multiplication shortcuts.
03:31 Let's collect all the terms.
03:38 Isolate B now, shall we?
03:42 This time, B meets the defined domain.
03:46 And that gives us the solution to our problem!

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Substitute the given solution x=−1 x = -1 into the equation.
  • Step 2: Simplify the equation and solve for the missing coefficient.

Now, let's work through each step:

Step 1: Substitute x=−1 x = -1 into the equation 7x2+□x+2=0 7x^2 + \square x + 2 = 0 .
This gives us:
7(−1)2+□(−1)+2=07(-1)^2 + \square(-1) + 2 = 0.

Step 2: Simplify the equation.
We know that (−1)2=1 (-1)^2 = 1 , so:
7×1−□+2=07 \times 1 - \square + 2 = 0.

This simplifies to:
7−□+2=07 - \square + 2 = 0,
which simplifies further to:
9−□=09 - \square = 0.

Solving for the blank, we have:
□=9\square = 9.

Therefore, the missing coefficient is 9 9 .

Answer

9

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