> [!summary] An algebraic proof is a type of proof where you show that LHS = RHS using algebra and manipulation. >[!info]+ Read Time **⏱ 1 min** # Definition An algebraic proof is a type of proof that relies on algebraic identities and manipulation (factoring, simplifying, [[Greatest Common Divisor (gcd)|greatest common divisors]], etc)to show that expressions are equal (Left hand side = Right hand side). # Examples > [!example] Show that $(a+b)^2 = a^2 + 2ab + b^2$ > > $ \begin{align*} (a+b)^2 &= (a+b)(a+b) \\ &= a\cdot a +a\cdot b+b\cdot a + b\cdot b \\ &= a^2 + 2ab + b^2 \end{align*} > $ > Therefore, using algebraic proof, LS = RS > [!example] Prove algebraically that the sum of two consecutive integers is always odd. > > Let any integer be $n$ Two consecutive integers are $n + n+1$ The sum of these is $2n +1$ This is the identity of a [[Even & Odd Numbers|odd number]] --- <!-- Light Mode Newsletter Embed --> <div class="mm-form-light"> <iframe src="https://updates.cyberleadhub.com/widget/form/Y0kpQVpjJQuxEfX59m17" id="inline-Y0kpQVpjJQuxEfX59m17" title="Join Math & Matter Newsletter (Light)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Dark Mode Newsletter Embed --> <div class="mm-form-dark"> <iframe src="https://updates.cyberleadhub.com/widget/form/lbeDLm24VjuaFxhjccA1" id="inline-lbeDLm24VjuaFxhjccA1" title="Join Math & Matter Newsletter (Dark)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Provider script (only once) --> <script src="https://updates.cyberleadhub.com/js/form_embed.js"></script>