Quadratic Expressions and Equations：練習問題

Factorize the following quadratic expression completely:
\( x^2 + 5x - 6 \)

Factorize the following quadratic expression completely:
\( x^2 - 12x + 36 \)

Factorize the following quadratic expression completely:
\( x^2 + 7x + 10 \)

Solve the quadratic equation \( 3x^2 - 12 = 0 \) by using the square root method.

A rectangle has a perimeter of \(20\) cm and its area is \(24\) cm\(^2\). If \(L\) and \(W\) are the length and width of the rectangle respectively, form a quadratic equation in terms of \(L\) and find the dimensions of the rectangle.

Consider the quadratic expression \(x^2 + 8x + 15\).
(a) Factorize the expression completely.
(b) Hence, solve the equation \(x^2 + 8x + 15 = 0\).

A rectangular garden has a length that is 3 meters more than its width. Let the width be \(x\) meters.
(a) Write an expression for the length of the garden in terms of \(x\).
(b) Given that the area of the garden is 40 m\(^2\), set up a quadratic equation in the form \(x^2 + bx + c = 0\).
(c) Solve the equation to find the dimensions of the garden. Explain why one of the solutions is rejected.