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 \(2x^2 – 5x – 12 = 0\).

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