Which equation has the same solution as $x^{2}-8 x-12=7$ ? $\newline$ $(x-4)^{2}=35$ $\newline$ $(x+4)^{2}=3$ $\newline$ $(x+4)^{2}=35$ $\newline$ $(x-4)^{2}=3$

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Grade 6

Factor numerical expressions using the distributive property

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Q. Which equation has the same solution as

x^{2}-8 x-12=7

\newline

(x-4)^{2}=35

\newline

(x+4)^{2}=3

\newline

(x+4)^{2}=35

\newline

(x-4)^{2}=3

Simplify Equation: Simplify the given equation by moving all terms to one side to set the equation to zero. $\newline$ $x^2 - 8x - 12 - 7 = 0$ $\newline$ $x^2 - 8x - 19 = 0$
Check First Choice: Look at the first choice $(x - 4)^2 = 35$ and expand it to see if it matches the simplified equation. $(x - 4)^2 = x^2 - 8x + 16$ This does not match the simplified equation $x^2 - 8x - 19$ because the constant term is different $(+16$ instead of $-19)$ .
Check Second Choice: Look at the second choice $(x + 4)^2 = 3$ and expand it to see if it matches the simplified equation. $(x + 4)^2 = x^2 + 8x + 16$ This does not match the simplified equation $x^2 - 8x - 19$ because the sign of the $x$ term is different $(+8x$ instead of $-8x)$ and the constant term is different $(+16$ instead of $-19)$ .
Check Third Choice: Look at the third choice $(x + 4)^2 = 35$ and expand it to see if it matches the simplified equation. $(x + 4)^2 = x^2 + 8x + 16$ This does not match the simplified equation $x^2 - 8x - 19$ because the sign of the $x$ term is different $(+8x$ instead of $-8x)$ and the constant term is different $(+16$ instead of $-19)$ .
Check Fourth Choice: Look at the fourth choice $(x - 4)^2 = 3$ and expand it to see if it matches the simplified equation. $(x - 4)^2 = x^2 - 8x + 16$ This does not match the simplified equation $x^2 - 8x - 19$ because the constant term is different $(+16$ instead of $-19)$ .
Adjust First Choice: Since none of the choices match the simplified equation $x^2 - 8x - 19$ when expanded, we need to check if any of the choices can be transformed to match the simplified equation by moving terms around. $\newline$ Let's add $19$ to both sides of the first choice to see if it matches the simplified equation: $\newline$ $(x - 4)^2 = 35$ $\newline$ $(x - 4)^2 + 19 = 35 + 19$ $\newline$ $x^2 - 8x + 16 + 19 = 54$ $\newline$ $x^2 - 8x + 35 = 54$ $\newline$ This does not match the simplified equation $x^2 - 8x - 19$ .
Adjust Second Choice: Add $19$ to both sides of the second choice to see if it matches the simplified equation: $\newline$ $(x + 4)^2 = 3$ $\newline$ $(x + 4)^2 + 19 = 3 + 19$ $\newline$ $x^2 + 8x + 16 + 19 = 22$ $\newline$ $x^2 + 8x + 35 = 22$ $\newline$ This does not match the simplified equation $x^2 - 8x - 19$ .
Adjust Third Choice: Add $19$ to both sides of the third choice to see if it matches the simplified equation: $\newline$ $(x + 4)^2 = 35$ $\newline$ $(x + 4)^2 + 19 = 35 + 19$ $\newline$ $x^2 + 8x + 16 + 19 = 54$ $\newline$ $x^2 + 8x + 35 = 54$ $\newline$ This does not match the simplified equation $x^2 - 8x - 19$ .
Adjust Fourth Choice: Add $19$ to both sides of the fourth choice to see if it matches the simplified equation: $\newline$ $(x - 4)^2 = 3$ $\newline$ $(x - 4)^2 + 19 = 3 + 19$ $\newline$ $x^2 - 8x + 16 + 19 = 22$ $\newline$ $x^2 - 8x + 35 = 22$ $\newline$ Subtract $22$ from both sides to get the simplified equation: $\newline$ $x^2 - 8x + 35 - 22 = 0$ $\newline$ $x^2 - 8x + 13 = 0$ $\newline$ This does not match the simplified equation $x^2 - 8x - 19$ .
Correct Approach: Realize that a mistake was made in the previous steps. The correct approach is to compare the given equation $x^2 - 8x - 12 = 7$ with the choices by first bringing the equation to the form $x^2 - 8x = 19$ (by adding $12$ to both sides and then subtracting $7$ from both sides). Then, we should look for a choice that, when simplified, has the form $x^2 - 8x = 19$ .