Solving Quadratic Equations

Solving by Factorising

How to Factorise a Quadratic Using the Area Model

Worked example

Factorise \(3x^2+10x-8\)

  1. Guide number: \((ax^2)(c)=(3x^2)(-8)=-24x^2\).
  2. Choose factors of the guide number that sum to 10x: \((-2x)+(12x)=10x\).
  3. Place \(3x^2,\ 12x,\ -2x,\ -8\) into the grid and take HCFs.

Hence \((3x-2)(x+4)\).

Solve \(3x^2+10x-8=0\)
\((3x-2)(x+4)=0 \Rightarrow x=\dfrac{2}{3}\) or \(x=-4\).
Procedure

Algorithm for \(ax^2+bx+c\)

  1. Compute the guide number \((ax^2)(c)=acx^2\).
  2. Find \(mx, nx\) with \(bx=mx+nx\) and \((mx)(nx)=(ax^2)(c)\).
  3. Arrange \(ax^2, mx, nx, c\) in a \(2\times2\) grid.
  4. Take HCFs across rows/columns to read the two binomial factors.
  5. If solving, set the product to zero and solve each factor.
Practice

Try these (factorise and solve)

  1. \(2x^2+7x+3=0\)
  2. \(4x^2-5x-6=0\)
  3. \(5x^2+9x-2=0\)
Show solutions

\(2x^2+7x+3=(2x+1)(x+3)\Rightarrow x=-\tfrac{1}{2},\,x=-3\).

\(4x^2-5x-6=(4x+3)(x-2)\Rightarrow x=2,\,x=-\tfrac{3}{4}\).

\(5x^2+9x-2=(5x-1)(x+2)\Rightarrow x=\tfrac{1}{5},\,x=-2\).

Using the Quadratic Formula

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula is used to solve equations of the form:

\[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants. The quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Example: Solve \( 2x^2 - 3x - 5 = 0 \)

We identify the coefficients:

\[ a = 2, \quad b = -3, \quad c = -5 \]

Step 1: Substitute the values into the quadratic formula

\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)} \]

Step 2: Evaluate both options using a calculator

Option 1 (Using \( + \) sign):

\[ x_1 = 2.5 \]

Option 2 (Using \( - \) sign):

\[ x_2 = -1 \]

Final Answer:

\[ x = 2.5 \quad \text{or} \quad x = -1 \]
Quadratic Coefficients Quiz

Quadratic Coefficients Quiz

When a quadratic equation is written as:

\( ax^2 + bx + c = 0 \)

The quadratic formula for solving it is:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this quiz, you will identify the values of coefficients \( a \), \( b \), and \( c \).