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n for all values of
x by completing the
sq

x^(2)+89=-20 x

$\mathrm{n}$ for all values of $\mathrm{x}$ by completing the $s q$ $\newline$ $x^{2}+89=-20 x$

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Composition of linear and quadratic functions: find a value

Full solution

Q.

\mathrm{n}

for all values of

\mathrm{x}

by completing the

s q

\newline

x^{2}+89=-20 x

Move Terms to One Side: First, let's move all terms to one side of the equation to set it equal to zero. $\newline$ $x^2 + 20x + 89 = 0$
Factor or Use Quadratic Formula: Now, we need to factor the quadratic equation, if possible. $\newline$ But this equation doesn't factor nicely, so we'll use the quadratic formula instead.
Calculate Discriminant: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = 20$ , and $c = 89$ .
Apply Quadratic Formula: Let's calculate the discriminant $b^2 - 4ac$ first. $\newline$ Discriminant = $20^2 - 4(1)(89)$ $\newline$ Discriminant = $400 - 356$ $\newline$ Discriminant = $44$
Simplify Square Root: Since the discriminant is positive, we have two real solutions. Now, plug the values into the quadratic formula. $x = \frac{-20 \pm \sqrt{44}}{2 \times 1}$
Divide by $2$ : Simplify the square root and the equation. $\newline$ $x = \frac{{-20 \pm \sqrt{{4 \cdot 11}}}}{2}$ $\newline$ $x = \frac{{-20 \pm 2\sqrt{11}}}{2}$
Divide by $2$ : Simplify the square root and the equation. $\newline$ $x = \frac{-20 \pm \sqrt{4 \times 11}}{2}$ $\newline$ $x = \frac{-20 \pm 2\sqrt{11}}{2}$ Divide all terms by $2$ to simplify further. $\newline$ $x = -10 \pm \sqrt{11}$

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