Solve the Quadratic Equation: x²+10x+25=0 Step-by-Step

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ a^2 + 2ab + b^2 $! In $ x^2 + 10x + 25 $, we have x² (first square), 25 (second square), and 10x (twice the product) since $ 2 \cdot x \cdot 5 = 10x $.

Q: Why does this equation have only one solution?

It's actually a double root! When $ (x + 5)^2 = 0 $, the factor $ (x + 5) $ appears twice, giving us $ x = -5 $ with multiplicity 2.

Q: Can I still use the quadratic formula?

Yes, but it's unnecessarily complicated! You'd get $ x = \frac{-10 \pm \sqrt{100-100}}{2} = \frac{-10 \pm 0}{2} = -5 $. Factoring is much faster!

Q: What if I don't see the perfect square pattern?

Practice recognizing common squares: 1, 4, 9, 16, 25, 36, 49... Then check if the middle term equals $ 2ab $. For $ x^2 + 10x + 25 $: does $ 10x = 2(x)(5) $? Yes!

Q: How do I verify my factoring is correct?

Expand your factored form! $ (x + 5)^2 = (x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25 $ ✓

Perfect Square Trinomials with Double Roots

Solve the following equation:

$x^2+10x+25=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Let's identify the coefficients

00:17 Let's use the roots formula

00:46 Let's substitute appropriate values according to the given data and solve

01:13 Let's calculate the square and products

01:24 The root of 0 is always equal to 0

01:29 When the root equals 0, there is only one solution to the equation

01:51 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:

$x^2+10x+25=0$

Step-by-step solution

The given quadratic equation is $x^2 + 10x + 25 = 0$ .

Notice that $x^2 + 10x + 25$ is a perfect square trinomial, which can be factored as $(x + 5)^2$ . Let's verify by expanding:

$(x + 5)^2 = (x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25$ .

Since the factoring is correct, we rewrite the equation:

$(x + 5)^2 = 0$ .

Take the square root of both sides:

$x + 5 = 0$ .

Solving for $x$ , we find:

$x = -5$ .

The equation has a double root, meaning the solution is repeated. Thus, the final solution is:

$x = -5$ .

This matches the given correct answer.

Final Answer

$x=-5$

Key Points to Remember

Essential concepts to master this topic

Recognition: Perfect square trinomial follows pattern $a^2 + 2ab + b^2$
Technique: Factor as $(x + 5)^2$ then solve $x + 5 = 0$
Check: Substitute $x = -5$ : $(-5)^2 + 10(-5) + 25 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Using quadratic formula unnecessarily
Don't automatically use the quadratic formula for $x^2 + 10x + 25 = 0$ = complex calculations with same result! This wastes time and increases error chances. Always check first if the trinomial is a perfect square that factors easily as $(x + 5)^2$ .

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

How do I recognize a perfect square trinomial?

Look for the pattern $a^2 + 2ab + b^2$ ! In $x^2 + 10x + 25$ , we have x² (first square), 25 (second square), and 10x (twice the product) since $2 \cdot x \cdot 5 = 10x$ .

Why does this equation have only one solution?

It's actually a double root! When $(x + 5)^2 = 0$ , the factor $(x + 5)$ appears twice, giving us $x = -5$ with multiplicity 2.

Can I still use the quadratic formula?

Yes, but it's unnecessarily complicated! You'd get $x = \frac{-10 \pm \sqrt{100-100}}{2} = \frac{-10 \pm 0}{2} = -5$ . Factoring is much faster!

What if I don't see the perfect square pattern?

Practice recognizing common squares: 1, 4, 9, 16, 25, 36, 49... Then check if the middle term equals $2ab$ . For $x^2 + 10x + 25$ : does $10x = 2(x)(5)$ ? Yes!

How do I verify my factoring is correct?

Expand your factored form! $(x + 5)^2 = (x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25$ ✓

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