Quadratics Core: quick revision, 19 bites

The Standard Form

The universal shape of every quadratic.

A **quadratic equation** is any polynomial equation of degree 2. Its standard form is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The coefficient $a$ is the **leading coefficient**, $b$ is the **linear coefficient**, and $c$ is the **constant term**.

Takeaway. Always rearrange a quadratic into $ax^2 + bx + c = 0$ before solving.

Roots of the Equation

Solutions are called roots.

The **solutions** or **roots** of a quadratic equation are the values of $x$ that make the equation true. Graphically, these are the points where the parabola $y = ax^2 + bx + c$ crosses the $x$-axis. A quadratic equation can have **two real roots**, **one real root** (a repeated root), or **two complex roots**.

Takeaway. Roots are the x-intercepts of the parabola.

Solving by Factoring

Break it down to zero.

Takeaway. Factoring works best when the equation has simple integer roots.

Factoring in Action

A concrete factoring example.

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

Takeaway. Check your factors by expanding them back to the original equation.

The Quadratic Formula

The universal solver.

The **quadratic formula** solves any quadratic equation, even when factoring is difficult or impossible. For $ax^2 + bx + c = 0$, the roots are given by:

Takeaway. Memorize the quadratic formula — it always works.

Using the Quadratic Formula

Plug, simplify, solve.

Takeaway. Be careful with signs when substituting negative coefficients.

Quadratic Formula Example

Solving with the formula.

Solve $2x^2 - 4x - 3 = 0$.

Takeaway. Always simplify the radical and the fraction completely.

The Discriminant

Predict roots without solving.

The **discriminant**, denoted $\Delta = b^2 - 4ac$, is the part of the quadratic formula under the square root. It tells you the nature of the roots **without** solving the equation.

Takeaway. Check the discriminant first to know what kind of solutions to expect.

Solving by Completing the Square

Make a perfect square trinomial.

Takeaway. Completing the square is the foundation for deriving the quadratic formula.

The Vertex Form

The parabola's turning point.

A quadratic can also be written in **vertex form**: $y = a(x - h)^2 + k$, where $(h, k)$ is the **vertex** of the parabola. This form is useful for graphing and for finding the maximum or minimum value.

Takeaway. The vertex form directly reveals the parabola's vertex.

Factoring vs. Quadratic Formula

Choosing the right tool.

Takeaway. Try factoring first, but fall back on the quadratic formula when needed.

Forgetting to Set to Zero

Zero is essential for factoring.

A frequent error is trying to factor an equation that is **not** set equal to zero. The Zero Product Property only applies when the product equals zero. Always rearrange the equation into the standard form $ax^2 + bx + c = 0$ **before** attempting to factor.

Takeaway. Always rearrange to standard form before factoring.

Sign Errors in the Formula

Watch those negative signs.

When using the quadratic formula, sign errors are common. Pay close attention to the **negative sign** in front of $b$ and the sign of $4ac$ when $c$ is negative. For example, if $b = -5$, then $-b = 5$. If $c = -3$, then $-4ac = -4(a)(-3) = +12a$.

Takeaway. Double-check your substitution, especially with negative coefficients.

Check Your Answers

Verify roots by substitution.

After solving, **plug your roots back** into the original equation to verify they work. This catches arithmetic errors and sign mistakes. It only takes a few seconds and can save you from losing marks on an otherwise correct solution.

Takeaway. Always verify your solutions by substituting them back.

Remembering the Formula

A song for the formula.

To memorize the quadratic formula, sing it to the tune of "Pop Goes the Weasel": *"x equals negative b, plus or minus the square root, of b-squared minus four a c, all over two a."* This catchy tune helps lock the formula into memory.

Takeaway. Use the 'Pop Goes the Weasel' tune to memorize the quadratic formula.

Projectile Motion

Quadratics in the real world.

Quadratic equations model **projectile motion**. The height $h(t)$ of an object thrown upward is given by $h(t) = -16t^2 + v_0t + h_0$ (in feet) or $h(t) = -4.9t^2 + v_0t + h_0$ (in meters), where $v_0$ is the initial velocity and $h_0$ is the initial height. Solving $h(t) = 0$ gives the time the object hits the ground…

Takeaway. Quadratic equations describe the path of any thrown or launched object.

Ancient Origins

Quadratics are thousands of years old.

Quadratic equations were solved by **Babylonian mathematicians** as early as 2000 BCE. They used geometric methods to find solutions, long before algebraic notation was invented. The quadratic formula itself was formalized much later, in the 9th century by the Persian mathematician **Al-Khwarizmi**.

Takeaway. The quadratic formula has a rich history spanning over 4000 years.

The Parabola as a Bridge

Visualizing the quadratic.

Think of a quadratic equation as describing the shape of a **suspension bridge** cable. The cable hangs in a parabolic curve. The roots are where the cable meets the road (the x-axis), and the vertex is the lowest (or highest) point of the cable. This visual helps connect the algebra to a physical shape.

Takeaway. The parabola's shape helps visualize roots and the vertex.

Simplify Radicals

Always simplify your final answer.

When your answer contains a square root, **simplify it completely**. For example, $\sqrt{50}$ should be written as $5\sqrt{2}$. Also, rationalize denominators when necessary. A fully simplified answer is required for full credit on most exams.

Takeaway. Always simplify radicals and rationalize denominators in your final answer.