Quadratic Equations Cheat Sheet & Examples

Quick answer

A quadratic equation has degree two and can be solved by factoring, completing the square, graphing, or using the quadratic formula.

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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.

Understanding Quadratic Equations

Factoring works because of the zero product rule. When two quantities are multiplied and the answer is zero, at least one quantity must be zero. This rule turns one harder equation into two simpler linear equations.

Before factoring, put every term on one side so the other side is zero. Then check for a greatest common factor first. Many students miss this easy step and make the remaining expression harder than it needs to be.

After finding possible values, substitute each one into the original equation. This check catches sign errors and confirms that both values really work.

Not every quadratic factors neatly using whole numbers. Completing the square is useful because it rewrites an expression into a form that shows its turning point clearly. The main idea is to create a binomial multiplied by itself.

For an expression with an x term, take half the coefficient of x, then square that result. Adding this same amount keeps the equation balanced only when it is handled on both sides.

This method can feel slow at first, but it explains where the general solving rule comes from. It is especially helpful when a graphing problem asks for the highest or lowest value rather than only the solutions.

The graph gives a visual meaning to the answers. A solution of the equation is an x value where the curve meets the horizontal axis. If the curve crosses that axis twice, there are two different real solutions.

If it only touches the axis at its turning point, the same solution occurs twice. If the entire curve stays above or below the axis, there is no real x value that makes the output zero. The coefficient on the squared term controls more than direction.

Its size affects the width of the curve. A larger absolute value makes the curve narrower, while a value closer to zero makes it wider.

Quadratics appear when a quantity rises then falls, or when an area depends on two changing lengths. In a projectile model, time is often the input and height is the output. The turning point represents the greatest height.

The zeros can represent when an object reaches the ground, though a negative time may not make sense in the situation. In area problems, roots can represent possible side lengths, but negative lengths must be rejected. Pay close attention to what each variable means and include units in a final statement.

Use a graph, a table, or substitution to check whether an answer fits the context. A correct calculation can still lead to an incorrect conclusion if an unrealistic value is kept.

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Quadratic Equations Cheat Sheet

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