Forming Quadratic Equations Revision Notes for Leaving Cert Mathematics

Forming Quadratic Equations

Understanding the process

When we solve quadratic equations, we find the roots (solutions) of the equation. Forming quadratic equations is the reverse process - we start with known roots and work backwards to create the equation.

The key principle

If we know the roots of a quadratic equation, we can form the equation using this fundamental relationship:

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If the roots are $a$ and $b$ , then the quadratic equation is $(x - a)(x - b) = 0$

Notice how we change the sign of each root when writing the factors.

Method for whole number roots

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Steps for whole number roots:

Identify the roots given in the question
Write the factors by changing the sign of each root
Set the product equal to zero: $(x - \text{root}_1)(x - \text{root}_2) = 0$
Expand to get the standard form

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Worked Example: Roots -4 and 5

Step 1: The roots are $-4$ and $5$

Step 2: Change signs to get factors: $(x - (-4))$ and $(x - 5)$ This gives us: $(x + 4)$ and $(x - 5)$

Step 3: Set equal to zero: $(x + 4)(x - 5) = 0$

Step 4: Expand using FOIL:

$x^2 - 5x + 4x - 20 = 0$
$x^2 - x - 20 = 0$

This is the quadratic equation with roots $-4$ and $5$ .

Method for fractional roots

Fractional roots require extra care. The key is to eliminate the fraction from the factor.

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Rule for fractional roots: If $x = \frac{a}{b}$ is a root, then $(bx - a)$ is the corresponding factor.

Understanding fractional roots

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How to handle fractional roots:

If $x = \frac{1}{2}$ is a root, then:

Multiply both sides by 2: $2x = 1$
Rearrange: $2x - 1 = 0$
Therefore: $(2x - 1)$ is the factor

Similarly, if $x = -\frac{1}{3}$ is a root:

Multiply both sides by 3: $3x = -1$
Rearrange: $3x + 1 = 0$
Therefore: $(3x + 1)$ is the factor

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Worked Example: Roots $-\frac{1}{4}$ and $3$

Step 1: The roots are $-\frac{1}{4}$ and $3$

Step 2: Form the factors:

For $x = -\frac{1}{4}$ : multiply by 4 to get $4x = -1$ , so factor is $(4x + 1)$
For $x = 3$ : factor is $(x - 3)$

Step 3: Set equal to zero: $(4x + 1)(x - 3) = 0$

Step 4: Expand:

$4x^2 - 12x + x - 3 = 0$
$4x^2 - 11x - 3 = 0$

This is the quadratic equation with roots $-\frac{1}{4}$ and $3$ .

Exam tips and common mistakes

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Critical points to avoid mistakes:

Always check your signs when forming factors from roots
Don't forget to multiply through when dealing with fractional roots
Expand carefully - use FOIL or the grid method to avoid errors
Write your final answer in standard form $(ax^2 + bx + c = 0)$
Verify your answer by checking that the original roots satisfy your equation

Quick check method

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Verification technique: To verify your equation is correct, substitute each root back into your final equation. If both roots make the equation equal zero, you've formed it correctly.

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Key Points to Remember:

Roots to factors: Change the sign of each root and put it in brackets with x
Fractional roots: Clear the fraction by multiplying through by the denominator
Always expand your factored form to get the standard quadratic equation
Check your work by substituting the original roots back into your equation
Practice is essential - this concept appears regularly in Leaving Cert exams

Forming Quadratic Equations (Leaving Cert Mathematics): Revision Notes