Full solution

Q. Solve for

x

. Enter the solutions from least to greatest.

\newline

2 x^{2}-24 x+54=0

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lesser

x=

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greater

x=

Write quadratic equation: Write down the quadratic equation. $\newline$ We have the quadratic equation $2x^2 - 24x + 54 = 0$ .
Factor quadratic equation: Factor the quadratic equation. $\newline$ To factor the quadratic equation, we need to find two numbers that multiply to give the product of the coefficient of $x^2$ (which is $2$ ) and the constant term (which is $54$ ), and add up to the coefficient of $x$ (which is $-24$ ). $\newline$ The product of the coefficient of $x^2$ and the constant term is $2 \times 54 = 108$ . $\newline$ We need two numbers that multiply to $108$ and add up to $-24$ . $\newline$ The numbers $-12$ and $2$ $0$ satisfy these conditions because $2$ $1$ and $2$ $2$ .
Rewrite using found numbers: Rewrite the quadratic equation using the numbers found. $\newline$ We can rewrite the middle term $-24x$ using the numbers $-12$ and $-9$ . $\newline$ $2x^2 - 24x + 54 = 2x^2 - 12x - 9x + 54$
Factor by grouping: Factor by grouping. $\newline$ Now we group the terms to factor by grouping: $\newline$ $(2x^2 - 12x) + (-9x + 54) = 0$ $\newline$ We can factor out a common factor of $2x$ from the first group and $-9$ from the second group: $\newline$ $2x(x - 6) - 9(x - 6) = 0$
Factor out common binomial: Factor out the common binomial factor. $\newline$ We notice that $(x - 6)$ is a common factor: $\newline$ $(2x - 9)(x - 6) = 0$
Solve for x: Solve for x using the zero product property. $\newline$ We set each factor equal to zero and solve for x: $\newline$ $2x - 9 = 0$ or $x - 6 = 0$ $\newline$ For $2x - 9 = 0$ , add $9$ to both sides to get $2x = 9$ , then divide by $2$ to get $x = \frac{9}{2}$ . $\newline$ For $x - 6 = 0$ , add $6$ to both sides to get $x = 6$ .
Write solutions in ascending order: Write the solutions in ascending order. $\newline$ The solutions are $x = \frac{9}{2}$ and $x = 6$ . In ascending order, we have $x = \frac{9}{2}$ , $x = 6$ .