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

Q: Why do I substitute the solution into the equation?

If $ x = -1 $ is truly a solution, then substituting it must make the equation equal zero. This creates a simple equation with only one unknown - the missing coefficient!

Q: What does the square symbol represent?

The square symbol (□) represents the unknown coefficient of the $ x $ term. It's just like a variable that we need to solve for, similar to solving for $ y $ or $ z $.

Q: Can a quadratic have more than one missing coefficient?

Yes, but you need at least as many known solutions as unknown coefficients. With one given solution, you can only find one missing coefficient.

Quadratic Equations with Unknown Coefficients

Look at the following equation:

$7x^2+\square x+2=0$

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following equation:

$7x^2+\square x+2=0$

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

Step-by-step solution

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

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

Now, let's work through each step:

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

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

This simplifies to:
$7 - \square + 2 = 0$ ,
which simplifies further to:
$9 - \square = 0$ .

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

Therefore, the missing coefficient is $9$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Substitution Rule: Replace variable with given solution value in equation
Technique: Substitute $x = -1$ gives $7(1) - \square + 2 = 0$
Check: Verify $7(-1)^2 + 9(-1) + 2 = 0$ equals zero ✓

Common Mistakes

Avoid these frequent errors

Forgetting to square the negative value correctly
Don't calculate $(-1)^2 = -1$ = wrong coefficient! This gives $-7 - \square + 2 = 0$ instead of $7 - \square + 2 = 0$ . Always remember that any number squared is positive: $(-1)^2 = 1$ .

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why do I substitute the solution into the equation?

If $x = -1$ is truly a solution, then substituting it must make the equation equal zero. This creates a simple equation with only one unknown - the missing coefficient!

What does the square symbol represent?

The square symbol (□) represents the unknown coefficient of the $x$ term. It's just like a variable that we need to solve for, similar to solving for $y$ or $z$ .

How do I know if my coefficient is correct?

Substitute your coefficient back into the original equation along with $x = -1$ . If you get $0 = 0$ , your answer is correct!

Can a quadratic have more than one missing coefficient?

Yes, but you need at least as many known solutions as unknown coefficients. With one given solution, you can only find one missing coefficient.

What if I get a negative coefficient?

Negative coefficients are perfectly valid! The sign depends on the numbers in your specific equation. Always double-check by substituting back to verify.

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