Quadratic Basics: quick revision, 15 bites

What is a Quadratic Equation?

The essential definition you must know.

A **quadratic equation** in the variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, $c$ are real numbers and $a \neq 0$. The term $ax^2$ is the **quadratic term**, $bx$ is the **linear term**, and $c$ is the **constant term**.

Takeaway. A quadratic equation is a second-degree polynomial equation set to zero.

Standard Form of a Quadratic

The universal format for all quadratics.

Any quadratic equation can be rewritten in this **standard form** by moving all terms to one side and simplifying. This form is the starting point for solving by factorization, completing the square, or using the quadratic formula.

Takeaway. Always rewrite a quadratic equation in the form $ax^2 + bx + c = 0$ before solving.

Identifying Coefficients

Spot a, b, and c quickly.

For the equation $2x^2 - 3x + 5 = 0$, the coefficients are:

Takeaway. Always note the signs: $b$ and $c$ include their signs.

Roots of a Quadratic Equation

The values that satisfy the equation.

A **root** (or solution) of a quadratic equation $ax^2 + bx + c = 0$ is a real number $\alpha$ such that $a\alpha^2 + b\alpha + c = 0$. In other words, substituting the root for $x$ makes the equation true. A quadratic equation can have at most two distinct roots.

Takeaway. Roots are the x-values where the quadratic expression equals zero.

Solving by Factorization

Step-by-step method to find roots.

Takeaway. Factorization converts a quadratic into two simple linear equations.

Factorization Example

See the method in action.

Solve $x^2 - 5x + 6 = 0$ by factorization.

Takeaway. For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$.

Forgetting to Set to Zero

A classic error that costs marks.

When solving an equation like $x^2 = 3x$, students often divide both sides by $x$ to get $x = 3$, losing the root $x = 0$. Always bring all terms to one side first: $x^2 - 3x = 0$, then factor: $x(x - 3) = 0$, giving $x = 0$ or $x = 3$.

Takeaway. Never divide by a variable that could be zero; factor instead.

The Quadratic Formula

A universal solver for any quadratic.

This formula gives the roots directly, even when factorization is not easy. The expression under the square root, $b^2 - 4ac$, is called the **discriminant**.

Takeaway. The quadratic formula works for every quadratic equation.

The Discriminant and Nature of Roots

Predict roots without solving.

For $ax^2 + bx + c = 0$, the discriminant $D = b^2 - 4ac$ determines the nature of the roots:

Takeaway. The discriminant tells you how many real roots exist.

Using the Quadratic Formula

Apply the formula step by step.

Solve $2x^2 - 4x - 6 = 0$ using the quadratic formula.

Takeaway. Substitute carefully and simplify the fraction.

Factorization vs. Quadratic Formula

Which method to choose?

Takeaway. Use factorization for simple quadratics; otherwise, use the formula.

Where Quadratics Appear

Real-life applications of quadratics.

Quadratic equations model **projectile motion** (height of a thrown object over time), **area problems** (finding dimensions of a rectangle given area), and **profit maximization** in business. For example, the height $h(t)$ of a ball thrown upward is often $h(t) = -4.9t^2 + v_0 t + h_0$.

Takeaway. Quadratics describe many natural and economic phenomena.

Check Your Roots

Verify your answers quickly.

After solving, substitute each root back into the original equation to ensure it gives $0$. This catches arithmetic errors and confirms you didn't miss a root.

Takeaway. Always verify roots by substitution.

Remember the Quadratic Formula

A catchy tune to memorize.

Sing the quadratic formula to the tune of *"Pop Goes the Weasel"*: *"x equals negative b, plus or minus the square root, of b-squared minus 4ac, all over 2a."*

Takeaway. Music helps memorize the formula.

Ancient Origins

Quadratics are thousands of years old.

Babylonian mathematicians solved quadratic-type problems as early as 2000 BC, using geometric methods. The general solution we use today was formalized much later.

Takeaway. Quadratics have a long history in mathematics.