Mathematics » Equations and Inequalities » Finding The Equation

# Finding the Equation

## Finding the Equation

We have seen that the roots are the solutions obtained from solving a quadratic equation. Given the roots, we are also able to work backwards to determine the original quadratic equation.

## Example

### Question

Find an equation with roots $$\text{13}$$ and $$-\text{5}$$.

### Assign a variable and write roots as two equations

$x = 13 \text{ or } x = -5$

Use additive inverses to get zero on the right-hand sides $x – 13 = 0 \text{ or } x + 5 = 0$

### Write down as the product of two factors

$(x-13)(x+5) = 0$

Notice that the signs in the brackets are opposite of the given roots.

### Expand the brackets

$x^2 – 8x – 65 = 0$

Note that if each term in the equation is multiplied by a constant then there could be other possible equations which would have the same roots. For example,

Multiply by $$\text{2}$$: $2x^2 – 16x – 130 = 0$

Multiply by $$-\text{3}$$: $-3x^2 + 24x + 195 = 0$

## Example

### Question

Find an equation with roots $$-\cfrac{3}{2}$$ and $$\text{4}$$.

### Assign a variable and write roots as two equations

$x = 4 \text{ or } x = -\cfrac{3}{2}$

Use additive inverses to get zero on the right-hand sides. $x – 4 = 0 \text{ or } x + \cfrac{3}{2} = 0$

Multiply the second equation through by $$\text{2}$$ to remove the fraction. $x – 4 = 0 \text{ or } 2x + 3 = 0$

### Write down as the product of two factors

$(2x+3)(x-4) = 0$

### Expand the brackets

The quadratic equation is $$2x^2 – 5x – 12 = 0$$.