Solve f(x) = x²: Finding the Input Value When f(?) = 49

Q: Why are there two answers when I solve x² = 49?

Because both 7 and -7 give the same result when squared! Remember: $ 7^2 = 49 $ and $ (-7)^2 = 49 $. This is why quadratic equations usually have two solutions.

Q: Do I always need to include the ± symbol?

Yes, when solving equations like x² = k! The ± symbol reminds you that both positive and negative values work. It's a crucial step that prevents you from missing solutions.

Q: What does 'Answers a + b' mean in the choices?

This means both answer choices a and b are correct. Since we found two solutions (x = 7 and x = -7), both $ f(7) $ and $ f(-7) $ equal 49.

Q: How do I verify my solutions are correct?

Substitute each value back into the original function: $ f(7) = 7^2 = 49 ✓ $ and $ f(-7) = (-7)^2 = 49 ✓ $. If both equal 49, you're right!

Q: Why isn't f(-3) a correct answer?

Because $ f(-3) = (-3)^2 = 9 $, not 49! Always double-check by substituting the value into the function to make sure it gives the desired output.

Quadratic Functions with Multiple Solutions

Complete:

The missing value of the function point:

$f(x)=x^2$

$f(?)=49$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Set up and solve

00:03 Let's substitute appropriate values according to the given data, and solve for X

00:12 Extract the root

00:15 When extracting a root there are 2 solutions, positive and negative

00:21 These are the 2 points

00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete:

The missing value of the function point:

$f(x)=x^2$

$f(?)=49$

Step-by-step solution

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

Step 1: Identify the given equation from the function.
Step 2: Use the square root method to find possible values of $x$ .
Step 3: Compare solutions with the provided answer choices.

Now, let's work through each step:
Step 1: Start with the equation $f(x) = 49$ which gives us $x^2 = 49$ .
Step 2: Solve for $x$ by taking the square root of both sides, which leads to two possible solutions: $x = \sqrt{49}$ and $x = -\sqrt{49}$ . Thus, $x = 7$ or $x = -7$ .
Step 3: Compare these solutions with the answer choices:

$f(7)$
$f(-7)$
$f(-3)$ (this is incorrect as $f(-3) = 9$ )
Answers a + b

The correct answers based on the solutions are

f(7)

and

f(-7)

, making choice 4, "Answers a + b," the correct option.

Therefore, the correct answer is: Answers a + b.

Final Answer

Answers a + b

Key Points to Remember

Essential concepts to master this topic

Rule: When solving x² = k, both positive and negative solutions exist
Technique: Use square root property: x = ±√49 gives x = 7 or x = -7
Check: Verify both solutions: (7)² = 49 and (-7)² = 49 ✓

Common Mistakes

Avoid these frequent errors

Finding only the positive solution
Don't solve x² = 49 as x = 7 only = missing half the answer! Squares of both positive and negative numbers are positive, so both solutions matter. Always write x = ±√49 to find both x = 7 and x = -7.

Practice Quiz

Test your knowledge with interactive questions

What is the value of y for the function?

$$ y=x^2 $$

of the point $$ x=2 $$ ?

FAQ

Everything you need to know about this question

Why are there two answers when I solve x² = 49?

Because both 7 and -7 give the same result when squared! Remember: $7^2 = 49$ and $(-7)^2 = 49$ . This is why quadratic equations usually have two solutions.

Do I always need to include the ± symbol?

Yes, when solving equations like x² = k! The ± symbol reminds you that both positive and negative values work. It's a crucial step that prevents you from missing solutions.

What does 'Answers a + b' mean in the choices?

This means both answer choices a and b are correct. Since we found two solutions (x = 7 and x = -7), both $f(7)$ and $f(-7)$ equal 49.

How do I verify my solutions are correct?

Substitute each value back into the original function: $f(7) = 7^2 = 49 ✓$ and $f(-7) = (-7)^2 = 49 ✓$ . If both equal 49, you're right!

Why isn't f(-3) a correct answer?

Because $f(-3) = (-3)^2 = 9$ , not 49! Always double-check by substituting the value into the function to make sure it gives the desired output.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime