Find the roots of the equation (2t + 3)² + (t + 4)² = 10....

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anakcianoalbiso @anakcianoalbiso

October 2022 1 5 Report

Find the roots of the equation (2t + 3)² + (t + 4)² = 10.

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✏️QUADRATIC

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- Find the roots of the equation (2t + 3)² + (t + 4)² = 10.

- Arrange the equation in standard form.

$\boxed{ \sf {ax}^{2} + bx + c = 0}$

${ \sf (2t + 3)^{2} + (t + 4)^{2} = 10}$

${ \sf (2t + 3)(2t + 3) + (t + 4)(t + 4) = 10}$

${ \sf (4t^{2} + 12t + 9) + ( {t}^{2} + 8t + 16) = 10}$

${ \sf 4t^{2} + {t}^{2} + 12t + 8t + 16 + 9 = 10}$

${ \sf 5 {t}^{2} + 20t + 25= 10}$

${ \sf 5 {t}^{2} + 20t + 25 - 10=0}$

${ \sf 5 {t}^{2} + 20t + 15=0}$

- Identify the values of a, b and c.

Values: a = 5, b = 20, c = 15

- Solve the equation using the quadratic formula.

$\boxed{\begin{gathered} \sf x = \frac{ - b \: ± \: \sqrt{{b}^{2} -4ac } }{2a} \end{gathered}}$

$\begin{gathered} \sf x = \frac{ -20 \: ± \: \sqrt{{20}^{2} -4(5)(15) } }{2(5)} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -20 \: ± \: \sqrt{400 -4(5)(15) } }{10} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -20 \: ± \: \sqrt{400 -4(75) } }{10} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -20 \: ± \: \sqrt{400 -300 } }{10} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -20 \: ± \: \sqrt{100} }{10} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -20 \: ± \: 10}{10} \end{gathered}$

Solution 1:

$\begin{gathered} \sf x = \frac{ -20 \: + \: 10}{10} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -10}{10} \end{gathered}$

$\begin{gathered} \sf x = - 1 \end{gathered}$

Solution 2:

$\begin{gathered} \sf x = \frac{ -20 \: - \: 10}{10} \end{gathered}$

$\begin{gathered} \sf x = \frac{ -30}{10} \end{gathered}$

$\begin{gathered} \sf x = - 3 \end{gathered}$

$\therefore\large \boxed{\sf \green{x = - 3,x = - 1}}$

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