Write a quadratic equation given the roots -1/3 and 5, show your work

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1. [tex] \boxed{(x – a)(x – b) = 0}[/tex]

   The equation above is the intercept form. Both a-term and b-term are the roots of equation.

   [tex]x = – \frac{1}{3} \\ x = 5[/tex]

   These are the roots of equation. Therefore we substitute a = – 1/3 and b = 5 in the equation.

   [tex](x + \frac{1}{3} )(x – 5) = 0[/tex]

   Here we can convert the expression x+1/3 to this.

   [tex]x + \frac{1}{3} = 0 \\ 3x + 1 = 0[/tex]

   Rewrite the equation.

   [tex](3x + 1)(x – 5) = 0[/tex]

   Simplify by multiplying both expressions.

   [tex]3 {x}^{2} – 15x + x – 5 = 0 \\ 3 {x}^{2} – 14x – 5 = 0[/tex]

   Answer Check

   Substitute the given roots in the equation.

   [tex]3 {(5)}^{2} – 14(5) – 5 = 0 \\ 75 – 70 – 5 = 0 \\ 75 – 75 = 0 \\ 0 = 0[/tex]

   [tex]3( – \frac{1}{3} )^{2} – 14( – \frac{1}{3}) – 5 = 0 \\ 3( \frac{1}{9} ) + \frac{14}{3} – 5 = 0 \\ \frac{1}{3} + \frac{14}{3} – \frac{15}{3} = 0 \\ \frac{15}{3} – \frac{15}{3} = 0 \\ 0 = 0[/tex]

   The equation is true for both roots.

   Answer

   [tex] \large \boxed {3 {x}^{2} – 14x – 5 = 0}[/tex]

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