Quadratic equations and roots — Mathematics, 14–17 years
How a squared unknown creates a curved relationship, and how its roots mark where that curve reaches zero.
A curved rule
A quadratic equation contains x², so its graph is usually a parabola rather than a straight line. Solving it means finding the x-values that make the expression equal zero; these are called roots or solutions. A parabola can cross the x-axis twice, touch it once, or miss it completely.
Why squares appear
Squares arise whenever area, products of changing lengths, or curved motion is involved. A rectangle with sides x and x + 3 has area x² + 3x, so questions about its area become quadratic equations. The equation is a way to reverse the process and recover the unknown length.
Factoring a quadratic
Solve x² − 5x + 6 = 0. Find two numbers whose product is 6 and whose sum is −5: they are −2 and −3. So x² − 5x + 6 becomes (x − 2)(x − 3). A product is zero when one factor is zero, giving x = 2 or x = 3. Substitution checks both answers.
The missing solution
People often find one root and stop, because linear equations normally have one answer. A quadratic can have two, so factor both parts and set each factor equal to zero. Also check the context: a negative length may solve the algebra but cannot describe a physical side.
Shapes and trajectories
Quadratic models can describe the path of a thrown ball, the shape of a satellite dish, or the area of a design that changes with one length. The model can locate a maximum height or a point where a quantity reaches zero. It is an approximation when real forces or shapes do not follow a perfect parabola.
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