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Solve for x. Enter the solutions from least to greatest. {:[4x^(2)+48 x+128=0],[" lesser "x=◻],[" greater "x=◻]:}

Solve for x x . Enter the solutions from least to greatest.\newline4x2+48x+128=0 4 x^{2}+48 x+128=0 \newlinelesser x= x= \newlinegreater x= x=

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Q. Solve for x x . Enter the solutions from least to greatest.\newline4x2+48x+128=0 4 x^{2}+48 x+128=0 \newlinelesser x= x= \newlinegreater x= x=
  1. Identify the quadratic equation: Identify the quadratic equation to solve.\newlineThe given quadratic equation is 4x2+48x+128=04x^2 + 48x + 128 = 0.\newlineWe need to find the values of xx that satisfy this equation.
  2. Factor out the greatest common factor: Factor out the greatest common factor (GCF) if possible.\newlineThe GCF of the coefficients 44, 4848, and 128128 is 44. Let's factor it out.\newline4(x2+12x+32)=04(x^2 + 12x + 32) = 0
  3. Factor the quadratic expression: Factor the quadratic expression inside the parentheses.\newlineWe need to find two numbers that multiply to 3232 and add up to 1212.\newlineThe numbers 44 and 88 satisfy these conditions because 4×8=324 \times 8 = 32 and 4+8=124 + 8 = 12.\newlineSo, we can write the factored form as:\newline4(x+4)(x+8)=04(x + 4)(x + 8) = 0
  4. Solve for x using the zero product property: Solve for x using the zero product property.\newlineThe zero product property states that if a product of factors equals zero, then at least one of the factors must be zero.\newlineSo, we set each factor equal to zero and solve for x:\newlinex+4=0x + 4 = 0 or x+8=0x + 8 = 0
  5. Solve each equation for x: Solve each equation for x.\newlineFor x+4=0x + 4 = 0:\newlinex=−4x = -4\newlineFor x+8=0x + 8 = 0:\newlinex=−8x = -8\newlineThese are the two solutions for xx.
  6. Order the solutions from least to greatest: Order the solutions from least to greatest.\newlineThe solutions are −8-8 and −4-4. Ordered from least to greatest, we have:\newlinelesser x=−8x = -8\newlinegreater x=−4x = -4

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