Q. Solve for \(x\). \[x^2 + 10x + 25 = 0\]

Q: How do you factor x^2 + 10x + 25?.

Factor as (x+5)^2 because 5+5=10 and 5\cdot5=25..

Q: What are the solutions of x^2 + 10x + 25 = 0?

The only solution is x=-5, a repeated (double) root from (x+5)^2=0..

Q: How does the quadratic formula apply here?

Discriminant b^2-4ac=100-100=0, so one real root: x=-b/(2a)=-10/2=-5.

Q: What does multiplicity 2 mean for the root x=-5?

The root occurs twice; the parabola touches the x-axis at (-5, 0) but does not cross it.

Q: What is the graph (vertex form) of the quadratic?

In vertex form y=(x+5)^2. Vertex at (-5,0); opens upward with axis of symmetry x=-5.

Q: Are there complex roots?

No: discriminant is zero, so the single real root x=-5 has multiplicity 2; no distinct complex roots.

Q: How do I check my solution?

Substitute x=-5: (-5)^2+10(-5)+25=25-50+25=0. It satisfies the equation.

Answer

\[
x^2+10x+25=(x+5)^2=0
\]
Hence \(x+5=0\), so
\[
x=-5
\]
(Double root)

Detailed Explanation

Problem

Solve for \[x\] in the equation

\[x^2 + 10x + 25 = 0\]

Step-by-step solution (extremely detailed)

Recognize the quadratic form and coefficients.The equation is a quadratic in standard form \[ax^2 + bx + c = 0\] with
\[a = 1,\; b = 10,\; c = 25\].
Check whether the quadratic is a perfect square trinomial.Compute the square of \[x + 5\] and compare:
\[(x + 5)^2 = x^2 + 2\cdot x \cdot 5 + 5^2 = x^2 + 10x + 25.\]

Because the expansion equals the left-hand side of the given equation, the quadratic is a perfect square.
Rewrite the equation using the perfect square factorization.Replace \[x^2 + 10x + 25\] by \[(x + 5)^2\] to get:
\[(x + 5)^2 = 0.\]
Solve the squared equation.Take the square root of both sides. Because the right-hand side is zero, the only possibility is:
\[x + 5 = 0.\]
Isolate x.Subtract 5 from both sides to find x:
\[x = -5.\]
Comment on multiplicity and verification (optional check).The factorization \[(x + 5)^2\] indicates a repeated root (root of multiplicity 2) at \[x = -5\].
Verify by substitution: replace x by -5 in the original expression:

\[(-5)^2 + 10(-5) + 25 = 25 – 50 + 25 = 0.\]

Thus the solution is confirmed.

Final answer

\[x = -5\]

See full solution

Solve math problems instantly with our AI homework help tools!

AI helper

Algebra FAQs

What kind of polynomial is \(x^2 + 10x + 25\)?

It is a quadratic (degree 2) and a perfect square trinomial since it equals \((x+5)^2\).

How do you factor \(x^2 + 10x + 25\)?.

Factor as \((x+5)^2\) because \(5+5=10\) and \(5\cdot5=25\)..

What are the solutions of \(x^2 + 10x + 25 = 0\)?

The only solution is \(x=-5\), a repeated (double) root from \((x+5)^2=0\)..

How does the quadratic formula apply here?

Discriminant \(b^2-4ac=100-100=0\), so one real root: \(x=-b/(2a)=-10/2=-5\).

What does multiplicity 2 mean for the root \(x=-5\)?

The root occurs twice; the parabola touches the x-axis at \((-5, 0)\) but does not cross it.

How can I solve it by completing the square?

What is the graph (vertex form) of the quadratic?

In vertex form \(y=(x+5)^2\). Vertex at \((-5,0)\); opens upward with axis of symmetry \(x=-5\).

Are there complex roots?

No: discriminant is zero, so the single real root \(x=-5\) has multiplicity 2; no distinct complex roots.

How do I check my solution?

Substitute \(x=-5\): \((-5)^2+10(-5)+25=25-50+25=0\). It satisfies the equation.

Solve quadratic equations with ease
Try our AI homework tools right now

252,312+ customers tried
Analytical, General, Biochemistry, etc.

Finance homework help Economics homework help Accounting homework help