Q. \(a^{2} + b^{2} =\).

Q: What identity relates a^2+b^2 to (a+b)^2?

a^2+b^2=(a+b)^2-2ab. This comes from expanding (a+b)^2=a^2+2ab+b^2 and rearranging.

Q: How do you minimize a^2+b^2 given a fixed sum a+b=s?

Minimized when a=b=\frac{s}{2}. Then a^2+b^2 = \frac{s^2}{2}, because a^2+b^2 = s^2-2ab and ab is maximal at \frac{s^2}{4}.

Q: What simple inequality involves a^2+b^2 and ab?

By (a-b)^2\ge0, we get a^2+b^2\ge 2ab. Equality holds when a=b.

Q: How is a^2+b^2 related to the Pythagorean theorem?

For integers, a^2+b^2=c^2 describes Pythagorean triples. Primitive triples are parameterized by integers m > n: a = k(m^2-n^2), b = k(2mn), c = k(m^2+n^2).

Q: When is an integer N expressible as a^2+b^2?

When is an integer N expressible as a^2+b^2?

Q: How can you view a^2+b^2 as a norm?

a^2+b^2 is the square of the Euclidean norm: \|(a,b)\|_2^2=a^2+b^2. Also |a+bi|^2=a^2+b^2 in \mathbb{C}.

Q: How do you complete the square for a^2+b^2?

Useful forms: a^2+b^2=(a+b)^2-2ab or a^2+b^2=(a-b)^2+2ab. Choose based on whether you want a sum or difference term.

Answer

\[
(a+b)^2=a^2+2ab+b^2\quad\Rightarrow\quad a^2+b^2=(a+b)^2-2ab
\]

Alternative form:
\[
(a-b)^2=a^2-2ab+b^2\quad\Rightarrow\quad a^2+b^2=(a-b)^2+2ab
\]

Detailed Explanation

Problem: Evaluate or express “a squared plus b squared”. In symbols: \(a^2 + b^2\).

Start from the square of a sum.

Compute the square of \(a + b\) by expanding the product \((a + b)(a + b)\). Use distributivity (FOIL):

\[
(a + b)^2 = a\cdot a + a\cdot b + b\cdot a + b\cdot b
\]

Combine like terms (\(a\cdot b\) and \(b\cdot a\) are equal):

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

Now solve this equation for \(a^2 + b^2\) by subtracting \(2ab\) from both sides:

\[
a^2 + b^2 = (a + b)^2 – 2ab
\]

Explanation: Expanding the square gives an expression that contains \(a^2 + b^2\) plus \(2ab\); isolating \(a^2 + b^2\) yields the identity above.
Alternative from the square of a difference.

Compute \((a – b)^2\) similarly:

\[
(a – b)^2 = a^2 – 2ab + b^2
\]

Solve for \(a^2 + b^2\) by adding \(2ab\) to both sides:

\[
a^2 + b^2 = (a – b)^2 + 2ab
\]

Explanation: This gives a second useful identity expressing the same quantity in terms of the square of the difference and the cross term \(2ab\).
Factorization over the complex numbers.

Over the real numbers \(a^2 + b^2\) does not factor into linear real factors (unless one of the variables is zero). Over the complex numbers it factors as:

\[
a^2 + b^2 = (a + bi)(a – bi)
\]

Verify by expanding the right-hand side:

\[
(a + bi)(a – bi) = a^2 – abi + abi – b^2 i^2 = a^2 + b^2
\]

Explanation: The imaginary terms cancel and \(i^2 = -1\) turns \(-b^2 i^2\) into \(+b^2\), recovering \(a^2 + b^2\).
Other useful rewritings (completing the square).

You can also rewrite \(a^2 + b^2\) by completing the square in different ways. For example:

\[
a^2 + b^2 = \left(a + \tfrac{b}{2}\right)^2 + \tfrac{3}{4}b^2
\]

Explanation: Expand \(\left(a + \tfrac{b}{2}\right)^2\) to get \(a^2 + ab + \tfrac{1}{4}b^2\); adding \(\tfrac{3}{4}b^2\) yields \(a^2 + b^2\).

Summary (compact forms):

\[
a^2 + b^2 = (a + b)^2 – 2ab = (a – b)^2 + 2ab = (a + bi)(a – bi)
\]

See full solution

Master every subject with Edubrain AI

Homework Helper

FAQs

What identity relates \(a^2+b^2\) to \((a+b)^2\)?

\(a^2+b^2=(a+b)^2-2ab\). This comes from expanding \((a+b)^2=a^2+2ab+b^2\) and rearranging.

Can \(a^2+b^2\) be factored?

Not over the real numbers into linear factors. Over the complex numbers \(a^2+b^2=(a+bi)(a-bi)\). Over reals you can factor as sum/difference of squares only with extra terms.

How do you minimize \(a^2+b^2\) given a fixed sum \(a+b=s\)?

Minimized when \(a=b=\frac{s}{2}\). Then \(a^2+b^2 = \frac{s^2}{2}\), because \(a^2+b^2 = s^2-2ab\) and \(ab\) is maximal at \(\frac{s^2}{4}\).

What simple inequality involves \(a^2+b^2\) and \(ab\)?

By \((a-b)^2\ge0\), we get \(a^2+b^2\ge 2ab\). Equality holds when \(a=b\).

How is \(a^2+b^2\) related to the Pythagorean theorem?

For integers, \(a^2+b^2=c^2\) describes Pythagorean triples. Primitive triples are parameterized by integers \(m > n\): \(a = k(m^2-n^2)\), \(b = k(2mn)\), \(c = k(m^2+n^2)\).

When is an integer \(N\) expressible as \(a^2+b^2\)?

How can you view \(a^2+b^2\) as a norm?

\(a^2+b^2\) is the square of the Euclidean norm: \(\|(a,b)\|_2^2=a^2+b^2\). Also \(|a+bi|^2=a^2+b^2\) in \(\mathbb{C}\).

How do you complete the square for \(a^2+b^2\)?

Useful forms: \(a^2+b^2=(a+b)^2-2ab\) or \(a^2+b^2=(a-b)^2+2ab\). Choose based on whether you want a sum or difference term.

Try three tools to solve a^2+b^2 now.
Pick a tool below.

185,791+ happy customers
Math, Calculus, Geometry, etc.

Pythagorean Theorem Calculator AI Math Solver Calculus AI