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$Solve for x. Enter the solutions from least to greatest. {:[5x^(2)+45 x+90=0],[" lesser "x=◻],[" greater "x=◻]:}$

Solve for $x$ . Enter the solutions from least to greatest. $\newline$ $\begin{array}{l} 5 x^{2}+45 x+90=0 \\ \text { lesser } x=\square \\ \text { greater } x=\square \end{array}$

Full solution

Q. Solve for

x

. Enter the solutions from least to greatest.

\newline

\begin{array}{l} 5 x^{2}+45 x+90=0 \\ \text { lesser } x=\square \\ \text { greater } x=\square \end{array}

Write Equation: Write down the given quadratic equation. $\newline$ $5x^2 + 45x + 90 = 0$ $\newline$ We need to find the values of $x$ that satisfy this equation.
Factor GCF: Factor out the greatest common factor (GCF) if possible. $\newline$ In this case, we can factor out a $5$ from each term. $\newline$ $5(x^2 + 9x + 18) = 0$
Factor Quadratic Expression: Now we need to factor the quadratic expression inside the parentheses. $\newline$ We are looking for two numbers that multiply to $18$ and add up to $9$ . $\newline$ The numbers $3$ and $6$ satisfy these conditions because $3 \times 6 = 18$ and $3 + 6 = 9$ . $\newline$ So we can write the factored form as: $\newline$ $5(x + 3)(x + 6) = 0$
Set Factors Equal to Zero: Set each factor equal to zero and solve for x. $\newline$ First, we set the factor outside the parentheses equal to zero: $\newline$ $5 = 0$ $\newline$ This does not give us a solution for x, so we ignore it. $\newline$ Next, we set each factor inside the parentheses equal to zero: $\newline$ $x + 3 = 0$ and $x + 6 = 0$
Solve Equations: Solve each equation for $x$ . $\newline$ For $x + 3 = 0$ : $\newline$ $x = -3$ $\newline$ For $x + 6 = 0$ : $\newline$ $x = -6$
Write Solutions: Write the solutions in ascending order. $\newline$ The lesser value of $x$ is $-6$ , and the greater value of $x$ is $-3$ .