Solve for all values of x : 3(2x+9)+(2x+9)^(2)=0 Answer: x=

Expand Squared Term: Simplify the equation by expanding the terms. We have the equation 3(2x+9)+(2x+9)^(2)=0. Let's expand the squared term first. (2x+9)^(2) = (2x+9)(2x+9)Expand Equation: Continue expanding the equation. (2x+9)(2x+9) = 4x^2 + 18x + 18x + 81 Combine like terms. 4x^2 + 36x + 81Substitute Expanded Term: Substitute the expanded squared term back into the original equation. 3(2x+9) + 4x^2 + 36x + 81 = 0 Expand the remaining term. 6x + 27 + 4x^2 + 36x + 81 = 0Combine Like Terms: Combine like terms. 4x^2 + 42x + 108 = 0Factor Quadratic Equation: Factor the quadratic equation. We look for two numbers that multiply to 4*108 and add up to 42. However, since 4*108 is 432 and 42 is not a factor of 432, we can't factor this easily. We should use the quadratic formula instead.Apply Quadratic Formula: Apply the quadratic formula. x = [-b ± sqrt(b^2 - 4ac)] / (2a) Here, a = 4, b = 42, and c = 108. x = [-42 ± sqrt(42^2 - 4*4*108)] / (2*4)Calculate Discriminant: Calculate the discriminant (b^2 - 4ac). Discriminant = 42^2 - 4*4*108 Discriminant = 1764 - 1728 Discriminant = 36Two Real Solutions: Since the discriminant is positive, there are two real solutions. x = [-42 ± sqrt(36)] / 8 x = [-42 ± 6] / 8Solve for x Values: Solve for the two values of x. First solution: x = (-42 + 6) / 8 x = -36 / 8 x = -4.5 Second solution: x = (-42 - 6) / 8 x = -48 / 8 x = -6

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Algebra 2

Composition of linear and quadratic functions: find a value

Full solution

Q. Solve for all values of

x

\newline

3(2 x+9)+(2 x+9)^{2}=0

\newline

Answer:

x=

Expand Squared Term: Simplify the equation by expanding the terms. $\newline$ We have the equation $3(2x+9)+(2x+9)^{2}=0$ . Let's expand the squared term first. $\newline$ $(2x+9)^{2} = (2x+9)(2x+9)$
Expand Equation: Continue expanding the equation. $\newline$ $(2x+9)(2x+9) = 4x^2 + 18x + 18x + 81$ $\newline$ Combine like terms. $\newline$ $4x^2 + 36x + 81$
Substitute Expanded Term: Substitute the expanded squared term back into the original equation. $\newline$ $3(2x+9) + 4x^2 + 36x + 81 = 0$ $\newline$ Expand the remaining term. $\newline$ $6x + 27 + 4x^2 + 36x + 81 = 0$
Combine Like Terms: Combine like terms. $\newline$ $4x^2 + 42x + 108 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ We look for two numbers that multiply to $4\times108$ and add up to $42$ . However, since $4\times108$ is $432$ and $42$ is not a factor of $432$ , we can't factor this easily. We should use the quadratic formula instead.
Apply Quadratic Formula: Apply the quadratic formula. $\newline$ $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $\newline$ Here, $a = 4$ , $b = 42$ , and $c = 108$ . $\newline$ $x = \frac{-42 \pm \sqrt{42^2 - 4\cdot4\cdot108}}{2\cdot4}$
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ . $\newline$ Discriminant = $42^2 - 4\times4\times108$ $\newline$ Discriminant = $1764 - 1728$ $\newline$ Discriminant = $36$
Two Real Solutions: Since the discriminant is positive, there are two real solutions. $\newline$ $x = \frac{-42 \pm \sqrt{36}}{8}$ $\newline$ $x = \frac{-42 \pm 6}{8}$
Solve for x Values: Solve for the two values of x. $\newline$ First solution: $\newline$ $x = (-42 + 6) / 8$ $\newline$ $x = -36 / 8$ $\newline$ $x = -4.5$ $\newline$ Second solution: $\newline$ $x = (-42 - 6) / 8$ $\newline$ $x = -48 / 8$ $\newline$ $x = -6$