Solve the Quadratic Equation: x² - 25 = 0 Step by Step

Q: Why are there two solutions to this equation?

Because when you square both 5 and -5, you get 25! Think about it: $ 5^2 = 25 $ and $ (-5)^2 = 25 $. So both values make the original equation true.

Q: Can I use factoring instead of square roots?

Yes! You can factor $ x^2 - 25 = 0 $ as $ (x-5)(x+5) = 0 $. This gives the same solutions: x = 5 and x = -5.

Q: What if the number under the square root isn't a perfect square?

You'd leave it in radical form! For example, if $ x^2 = 7 $, then $ x = ±\sqrt{7} $. You don't need to calculate the decimal unless specifically asked.

Q: How do I know this is a quadratic equation?

Look for the highest power of x! Since we have $ x^2 $ (x to the second power), this is a quadratic equation. The standard form is $ ax^2 + bx + c = 0 $.

Q: Why do we write the answer as x₁ = 5, x₂ = -5?

The subscripts help us distinguish between the two solutions. In math, we often label multiple answers this way. You could also write the solution set as $ \{-5, 5\} $.

Quadratic Equations with Perfect Square Forms

Solve the following equation

$x^2-25=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Isolate X

00:11 Extract root

00:16 When extracting a root there are always 2 solutions (positive, negative)

00:21 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

Solve the following equation

$x^2-25=0$

Step-by-step solution

To solve the equation $x^2 - 25 = 0$ , follow these steps:

Step 1: Isolate the square term: Begin by rewriting the equation to isolate $x^2$ :

$x^2 - 25 = 0$
Add $25$ to both sides to obtain:
$x^2 = 25$

Step 2: Extract the square roots: To solve $x^2 = 25$ , take the square root of both sides. Remember to consider both the positive and negative roots:

$x = \pm \sqrt{25}$

Step 3: Simplify: Calculate the square root of 25:

$\sqrt{25} = 5$

Therefore, the solutions are:
$x = 5$
and
$x = -5$

Thus, the solutions to the equation $x^2 - 25 = 0$ are $x_1 = 5$ and $x_2 = -5$ .

Verifying with the provided choices, the correct choice matches the solution $x_1 = 5, x_2 = -5$ .

Therefore, the solution to the problem is $x_1 = 5, x_2 = -5$ .

Final Answer

$x_1=5,x_2=-5$

Key Points to Remember

Essential concepts to master this topic

Rule: Move constant to right side before taking square roots
Technique: From $x^2 = 25$ , get $x = ±5$
Check: Substitute both solutions: $5^2 - 25 = 0$ and $(-5)^2 - 25 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative square root
Don't just write x = 5 when solving $x^2 = 25$ = missing half the solution! Every positive number has two square roots. Always write $x = ±\sqrt{25}$ to get both x = 5 and x = -5.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

Why are there two solutions to this equation?

Because when you square both 5 and -5, you get 25! Think about it: $5^2 = 25$ and $(-5)^2 = 25$ . So both values make the original equation true.

Can I use factoring instead of square roots?

Yes! You can factor $x^2 - 25 = 0$ as $(x-5)(x+5) = 0$ . This gives the same solutions: x = 5 and x = -5.

What if the number under the square root isn't a perfect square?

You'd leave it in radical form! For example, if $x^2 = 7$ , then $x = ±\sqrt{7}$ . You don't need to calculate the decimal unless specifically asked.

How do I know this is a quadratic equation?

Look for the highest power of x! Since we have $x^2$ (x to the second power), this is a quadratic equation. The standard form is $ax^2 + bx + c = 0$ .

Why do we write the answer as x₁ = 5, x₂ = -5?

The subscripts help us distinguish between the two solutions. In math, we often label multiple answers this way. You could also write the solution set as $\{-5, 5\}$ .

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