Quadratic Equations Mind Map

A quadratic polynomial is an expression of degree 2, while a quadratic equation is that expression written equal to zero.

If the question asks for roots, first check that the expression is written as an equation equal to zero.

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

Always compare the given equation term by term with ax^2 + bx + c = 0 before using any formula.

This means turning a word problem or geometry problem into a quadratic equation by choosing a variable and forming the correct relation.

Choose one variable clearly, build the relation from the question, and expand carefully before solving.

The roots of a quadratic equation are the values of x that make the equation true, meaning they make the left side equal to zero.

To check a root, substitute it back into the original equation and see whether the left side becomes zero.

Factorisation means rewriting a quadratic equation as a product of two simpler expressions and then using the zero product principle.

Check both the product and the sum while choosing factor pairs, especially for negative middle terms.

If the product of two expressions is zero, then at least one of the expressions must be zero.

After factorisation, never keep the product equal to zero only; split it into two separate equations.

A quadratic equation is any equation that can be simplified to ax^2 + bx + c = 0 with a ≠ 0.

Always simplify first. The equation is quadratic only if the highest power after simplification is exactly 2.

Completing the square means rewriting a quadratic expression so that part of it becomes a perfect square trinomial.

For x^2 + bx, add (b/2)^2 to complete the square, then keep the equation balanced.

The quadratic formula gives the roots of any quadratic equation ax^2 + bx + c = 0 as x = (-b ± √(b² - 4ac)) / 2a.

First write the equation in standard form, then copy a, b, c carefully before substituting into the formula.

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac.

Use the discriminant first when the question asks about the nature of roots.

The nature of roots depends on the discriminant D: if D > 0 the roots are real and distinct, if D = 0 they are real and equal, and if D < 0 they are not real.

Memorise the sign rule carefully: positive means distinct, zero means equal, negative means no real roots.

Repeated roots are two equal roots of a quadratic equation, and they occur when the discriminant is zero.

Whenever D = 0, state that the roots are real and equal, not just real.

If the discriminant of a quadratic equation is negative, the equation has no real roots.

If D is negative, state clearly that the equation has no real roots and stop there unless complex roots are asked.

Equation reduction means converting a formed relation into a simpler quadratic equation in standard form by expanding and rearranging terms.

Always end the reduction step with '= 0', because standard form makes solving easier.

Selecting a suitable method means choosing factorisation, completing the square, or the quadratic formula depending on the form of the equation.

If you cannot see a clean factor pair quickly, move to the quadratic formula instead of guessing.

Verification means substituting the obtained roots back into the original quadratic equation to check whether they really satisfy it.

When asked to verify, always substitute into the original equation, not into an already simplified version only.

These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation.

For a positive constant with a negative middle term, both signs inside the brackets are usually negative.

These are mistakes made while substituting a, b, and c into the quadratic formula, especially sign mistakes and denominator mistakes.

Write a, b, and c separately before starting the formula; do not substitute in a hurry.

This means forming a quadratic equation from area-based geometry information such as rectangles, squares, or borders.

Use a quick sketch for the shape, label the sides, and then write the area relation before expanding.

This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities.

For two numbers x and y, remember: sum becomes the x-term and product becomes the constant term after standard form is made.