Math Concepts Archives – Mathemafia https://mathemafia.com/category/math-concepts/ Mathematics is like music - Enjoy it!!! Sun, 30 Mar 2025 12:14:02 +0000 en-US hourly 1 https://wordpress.org/?v=7.1 https://mathemafia.com/wp-content/uploads/2020/08/cropped-M__Logo__SM-1-32x32.png Math Concepts Archives – Mathemafia https://mathemafia.com/category/math-concepts/ 32 32 Quadratic Equation Calculator https://mathemafia.com/quadratic-equation-calculator/ Wed, 26 Mar 2025 07:57:52 +0000 https://mathemafia.com/?p=15877 Quadratic Equation Calculator - A Complete solution of any quadratic polynomial equation that provides the step by step explanation of the finding the roots of the equation or zeros of equivalent quadratic polynomial.

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\( \begin{aligned} \end{aligned}\)

Quadratic Equation Calculator

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Enter the coefficeints of QE

Coeff. the x2

Coeff. the x

Constant

\( \begin{aligned}\text{Coeff of x}^2\text{ cannot be 0}\end{aligned}\)

Answer

Solution

Comparing the equation with
ax2 + bx + c = 0, we get

Discriminant (D)

Nature of Roots

D > 0 β‡’ Two Real and Distinct Roots

D = 0 β‡’ Two Real and Equal Roots

D < 0 β‡’ Two Complex Roots

Roots

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A polynomial equation of degree 2 is called Quadratic Equation. It’s standard form is

ax2 + bx + c = 0.

Roots of a Quadratic Equation

A quadratic equation has exactly 2 roots. These roots are either both Real or both Complex. It is not possible to have 1 real and 1 complex root for any quadratic equation with real coefficients.Β 

Click here to read about 1 real and 1 complex root scenario

Nature of Roots

To find the nature of roots, we need to find the discriminant of the quadratic equation. Formula to calculate the discriminant is:

\(\begin{aligned}&D = b^2-4ac\end{aligned}\)
D > 0 β‡’ Two Real and Distinct Roots

D = 0 β‡’ Two Real and Equal Roots

D < 0 β‡’ Two Complex Roots

It is always recommended to find the discriminant first rather than directly solving the equation.

Finding Roots

Once the discriminant is calculated, we can find the roots of the Quadratic Equation using the Quadratic Formula.

\(\begin{aligned}&x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\end{aligned}\)

An easy way to remember this formula is:

\(\begin{aligned}&x = \frac{-b \pm \sqrt{D}}{2a}\end{aligned}\)

Note: The roots of a quadratic equation are the zeros of equivalent quadratic polynomial.Β 

1 Real Root & 1 Complex Root Scenario

If we have a quadratic equation with imaginary or complex coefficients, then the equation can have 1 real and 1 complex root. For example, the roots of the equation mentioned below are: 3i and 1.Β 

\(\begin{aligned}&x^2-(3i + 1)x+(3i)=0\end{aligned}\)

An easy way to understand this is –

Form an equation like this:

\(\begin{aligned}&(x-3i) (x-1)=0\\\\&(x-\text{[A complex number]}) (x-\text{[A real number]})=0\end{aligned}\)

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]]> SSA is not a valid congruence criterion https://mathemafia.com/ssa-is-not-a-valid-congruence-criterion/ Sun, 09 Mar 2025 12:17:43 +0000 https://mathemafia.com/?p=13890 A witty explanation and proof by mathemafia that SSA is not a valid congruence criterion when it comes to the congruence of triangles in geometry.

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SSA is not a valid congruence criterion

Student: What is SSA?

Mathemafia – I call it Side Side Aaaaaaaaaa…..hhhhhhΒ !!!

Not funny, huh!! But effective 🎯 enough to remember 🧠.

If you want to read, it may take 5 minutes. If you want to save time, watch the short video and then send me half πŸŒ“ of the money you save.Β 

You know… ⏰ Time is πŸ’° Money πŸ˜‚ 🀣 πŸ˜†

If you still insist, written below is the explanation why SSA is not a valid congruence criterion. 🀿 πŸ‘€ ❀️ πŸ’ƒ

But First, Let’s Sing it!!!

As I always say – Mathematics is like Music. Enjoy it!!! But don’t enjoy this one alone πŸ₯³. This piece of music (SSA) is a party πŸŽ‰ song 🎢. So enjoy it with your friends. πŸ’ƒπŸ•ΊπŸ’ƒπŸ•Ί. (I can’t ask in more creative 🎭 way to share πŸ”— it. Hmmmmm… Let me say it clearly then… I BEG YOU. SHARE it pls. 😭 πŸ™ πŸ˜­πŸ™ )Β 

SSA – Sing Sing Aaaaaaaa…..Β 

SSA – Share Share Aaaaaaaa…..Β 

SSA – Subscribe Subscribe Aaaaaaaa…..Β 

StudentΒ – Sir, how about if we reverse the abbreviation 🀭 πŸ™ˆ πŸ˜† πŸ‘ 🫏

Mathemafia – Wow!!! That’s much more creative. 🫏 🫏 

Explanation

Let us draw an obtuse angled triangle, ABC with obtuse angle at vertex A as shown in the Pic 1.

Obviously side BC will be the longest side.Β 

Side opposite to largest angle is the longest side in a triangle.

Pic 1
Pic 2

Take a compass and draw an arc of length AB. It will intersect the longest side BC, say at point D.

See Pic 2.

Join AD.

Let us now focus on two trianglesβ–²ABC and β–²ADC

\( \begin{aligned} \overline{AB} &\cong \overline{AD} \ \ (Construction) \\ \overline{AC} &\cong \overline{AC} \ \ (Common) \\ \angle C &= \angle C \ \ (Common) \end{aligned}\)

So we got two congruent sides and one congruent angle in SSA formation but clearly the two triangles are not congruent. Hence, SSA is not a valid congruent criterion.Β 

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