Question

Factorise (a) \(14 x^{2}-127 x-57\) (b) \(45 x^{2}+44 x+7\), (c) \(6 x^{2}+19 x-11\).

Short answer

**Question**: Factorise the following quadratic expressions: (a) \(14x^2 - 127x - 57\) (b) \(45x^2 + 44x + 7\) (c) \(6x^{2}+19 x-11\) **Short Answer**: (a) \((x - 7)(x + 1)\) (b) \((5x + 4)(9x + 1)\) (c) \((3x+11)(2x-1)\)

Step-by-step solution

Identify the quadratic expression.

The expression is \(14x^2 - 127x - 57\).

Use the quadratic formula to find its factors

We'll use the quadratic formula to find the factors of the quadratic expression. The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Where \(a, b, c\) are coefficients of \(14x^2 - 127x - 57\). Here, \(a = 14, b = -127, c = -57\). Plugging these values in the quadratic formula: \(x = \frac{127 \pm \sqrt{(-127)^2 - (4)(14)(-57)}}{2(14)}\) \(x = \frac{127 \pm \sqrt{16129}}{28}\) Now we notice that \(16129=11 \cdot 147 \Rightarrow 127 \pm 441\) \(x_1 = \frac{127 + 441}{28} = \frac{568}{28} = 7x\) \(x_2 = \frac{127 - 441}{28} = \frac{-314}{28} = -x\) So the factors are \((x - 7)(x + 1)\) The factorised form of the expression is \((x - 7)(x + 1)\). (b) Factorise \(45x^2 + 44x + 7\)

Identify the quadratic expression.

The expression is \(45x^2 + 44x + 7\).

Factorise by grouping

Arrange the terms in different groups: \(45x^2 + 36x + 8x + 7\) Factor out the common factors of each group: \(9x(5x + 4) + 1(5x + 4)\) Factor out the common binomial factor \((5x + 4)\): \((5x + 4)(9x + 1)\) The factorised form of the expression is \((5x + 4)(9x + 1)\). (c) Factorise \(6x^{2}+19 x-11\)

Identify the quadratic expression.

The expression is \(6x^2 + 19x - 11\).

Factorise by grouping

Arrange the terms in different groups: \(6x^2 + 22x - 3x - 11\) Factor out the common factors of each group: \(2x(3x + 11) - 1(3x + 11)\) Factor out the common binomial factor \((3x + 11)\): \((3x+11)(2x-1)\) The factorised form of the expression is \((3x+11)(2x-1)\).

Key concepts

Quadratic Expression

A quadratic expression is a polynomial of degree 2. This means it takes the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients and \(a eq 0\). Quadratic expressions have distinct characteristics.
They graph as parabolas, which means the shape is U-shaped or an inverted U. The coefficient \(a\) determines the direction the parabola opens; if \(a > 0\), it opens upward, and if \(a < 0\), it opens downward.

• The squared term, \(ax^2\), dictates the parabola's width; larger values of \(a\) mean a narrower parabola.
• The linear term, \(bx\), affects the axis of symmetry, or how the parabola is positioned horizontally.
• The constant term, \(c\), represents the parabola's y-intercept, or the point at which it crosses the y-axis.

Understanding these features helps us to work with quadratic expressions effectively. Identifying the coefficients accurately is the first step in any form of factorization or solving process.

Quadratic Formula

The quadratic formula is a powerful tool that allows us to find the roots of any quadratic expression. Given a quadratic expression \(ax^2 + bx + c\), the quadratic formula is written as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula derives from the method of completing the square and is applicable in situations where factoring by other means isn't straightforward.

• \(b^2 - 4ac\) is called the discriminant; it determines the nature of the roots:
  • If the discriminant is positive, the quadratic has two distinct real roots.
  • If it is zero, the quadratic has exactly one real root (a perfect square trinomial).
  • If negative, the quadratic has two complex roots.
• The \(\pm\) symbol indicates the presence of two potential solutions, stemming from the square root operation.

By plugging in the values of \(a\), \(b\), and \(c\) from your quadratic expression, you can calculate these roots. After finding the roots, you can rewrite the expression as \((x - x_1)(x - x_2)\), where \(x_1\) and \(x_2\) are the roots. This process is crucial in factorization, transforming a seemingly complex quadratic into a product of linear factors.

Factoring by Grouping

Factoring by grouping is a useful technique used to factor polynomials, especially when direct factoring is difficult. It involves rearranging and grouping terms to reveal common factors.
The basic idea is to organize the terms such that you have pairs or groups that share a common factor, which can then be factored out to simplify the expression.To perform factoring by grouping:

• Divide the expression into two pairs of terms.
• Factor each pair individually to find the greatest common factor in each group.
• Look for a common binomial factor across the groups.
• Factor out the common binomial factor, resulting in a product of simpler expressions.

This method is versatile and widely used when dealing with complex quadratics, as it breaks them down into manageable parts. For example, in the expression \(6x^2 + 22x - 3x - 11\), rearranging and grouping the terms allows us to discover common factors and eventually factor them into \((3x+11)(2x-1)\). Employing factoring by grouping simplifies the expressions and often results in cleaner, more elegant solutions for quadratic expressions.