Quadratic Equations Cheat Sheet

A printable reference covering standard form, factoring, completing the square, the quadratic formula, discriminants, roots, and parabolas for grades 9-10.

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Quadratic equations are equations that include a squared variable and can model paths, areas, revenue, and many curved relationships. This cheat sheet helps students recognize quadratic form, choose a solving method, and connect equations to graphs. It is useful for checking steps quickly when solving, graphing, or interpreting answers. The main goal is to understand how roots, vertices, and coefficients work together. A quadratic equation is usually written as $ax^2+bx+c=0$ , where $a \neq 0$ . Students can solve quadratics by factoring, using square roots, completing the square, or applying the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ . The discriminant $b^2-4ac$ tells how many real solutions the equation has. The graph of $y=ax^2+bx+c$ is a parabola with vertex $\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$ .

Key Facts

Standard form for a quadratic equation is $ax^2+bx+c=0$ , where $a \neq 0$ .
The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for solving $ax^2+bx+c=0$ .
The discriminant is $D=b^2-4ac$ , and it determines the type and number of roots.
If $D>0$ , the quadratic has two distinct real roots; if $D=0$ , it has one repeated real root; if $D<0$ , it has no real roots.
A factorable quadratic can be solved by writing $ax^2+bx+c=(px+q)(rx+s)$ and setting each factor equal to $0$ .
For a perfect square equation, if $(x-h)^2=k$ , then $x=h\pm\sqrt{k}$ .
The axis of symmetry of $y=ax^2+bx+c$ is $x=\frac{-b}{2a}$ .
The vertex of $y=ax^2+bx+c$ is $\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$ , and the parabola opens up if $a>0$ and down if $a<0$ .

Vocabulary

Quadratic equation: A quadratic equation is an equation that can be written as $ax^2+bx+c=0$ , where $a \neq 0$ .
Root: A root is a value of $x$ that makes the equation $ax^2+bx+c=0$ true.
Parabola: A parabola is the U-shaped graph of a quadratic function such as $y=ax^2+bx+c$ .
Vertex: The vertex is the highest or lowest point of a parabola, located at $x=\frac{-b}{2a}$ for $y=ax^2+bx+c$ .
Discriminant: The discriminant is $D=b^2-4ac$ , the part of the quadratic formula that tells the number and type of roots.
Axis of symmetry: The axis of symmetry is the vertical line $x=\frac{-b}{2a}$ that divides a parabola into two matching halves.

Common Mistakes to Avoid

Forgetting that $a \neq 0$ is wrong because if $a=0$ , the equation is linear, not quadratic.
Dropping the $\pm$ in $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ is wrong because most quadratics with $D>0$ have two solutions.
Using $\frac{-b}{2a}$ as the only solution is wrong because it gives the axis of symmetry, not necessarily the roots.
Factoring without setting each factor equal to $0$ is wrong because $(x-r)(x-s)=0$ is solved by using the zero product property.
Taking the square root of both sides without both signs is wrong because $x^2=k$ gives $x=\pm\sqrt{k}$ when $k>0$ .

Practice Questions

1 Solve $x^2-5x+6=0$ by factoring.
2 Use the quadratic formula to solve $2x^2+3x-2=0$ .
3 Find the discriminant and the number of real roots for $3x^2-4x+7=0$ .
4 Explain how the signs of $a$ and $D=b^2-4ac$ affect the shape of the parabola and the number of $x$ -intercepts.