UPSKILL MATH PLUS

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Learn moreAbraham bar Hiyya Ha-Nasi, also known as Savasorda, was an Arab mathematician famous for his book "Liber Embadorum," published in \(1145 \ AD(CE)\) and was the first book in Europe to present the complete solution to the quadratic equation.

Quadratic expression

The polynomial is an expression of degree \(n\) for the variable \(x\) is of the form \(p(x) = a_0 x^n + a_{1}x^{n-1} + … + a_{n -1}x + a_n\), where \(a_0, a_1, a_2, …, a_n\) are coefficients and \(a_0 \ne 0\).

A quadratic expression is a polynomial of degree \(2\) for the variable \(x\) is of the form \(p(x) = ax^2 + bx + c\), \(a \ne 0\) and \(a\), \(b\) and \(c\) are real numbers.

Example:

**1**. Check \(p(x) = (x - 4)^3 + 4x + 2\) is a quadratic expression.

**Solution**:

\(p(x) = (x - 4)^3 - 4x + 2\)

\(=\) \(x^3 - 12x^2 + 48x - 64 - 4x + 2\) [Using the identity \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)]

\(=\) \(x^3 - 12x^2 + 44x - 62\)

Here, the degree of the equation is \(3\).

**Thus, the given equation is not a quadratic expression**.

**2**. Check \(p(x) = (x + 2)^3 - x^3 + 3\) is a quadratic expression.

**Solution**:

\(p(x) = (x + 2)^3 - x^3 + 3\)

\(=\) \(x^3 + 6x^2 + 12x + 8 - x^3 + 3\) [Using the identity \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)]

\(=\) \(6x^2 + 12x - 11\)

Here, the degree of the polynomial is \(2\).

**Thus, the given equation is a quadratic expression**.