Question
x^2-14+__=0 supply the number that has 6 and 8 as its solution
Answers
Answered by
GPT-4o mini
To find the number that can complete the equation \( x^2 - 14 + \_\_ = 0 \) so that 6 and 8 are the solutions, we can use the fact that if 6 and 8 are the roots of the equation, then we can express the quadratic in its factored form.
The roots 6 and 8 can be used to write the equation in the factored form:
\[
(x - 6)(x - 8) = 0
\]
Expanding this gives:
\[
x^2 - (6 + 8)x + (6 \cdot 8) = 0
\]
Calculating the coefficients:
\[
x^2 - 14x + 48 = 0
\]
To match this with the form \( x^2 - 14 + \_\_ = 0 \), we see that our equation has a -14x term and a constant term of 48.
To complete the original equation, we can rewrite it to find the constant term:
\[
x^2 - 14 + \_\_ = 0
\]
This suggests we solve for the blank that will satisfy the equation \( -14 + \_\_ = 48 \).
Thus, we find:
\[
\_\_ = 48 + 14 = 62
\]
Therefore, the number that completes the equation is:
\[
\boxed{62}
\]
The roots 6 and 8 can be used to write the equation in the factored form:
\[
(x - 6)(x - 8) = 0
\]
Expanding this gives:
\[
x^2 - (6 + 8)x + (6 \cdot 8) = 0
\]
Calculating the coefficients:
\[
x^2 - 14x + 48 = 0
\]
To match this with the form \( x^2 - 14 + \_\_ = 0 \), we see that our equation has a -14x term and a constant term of 48.
To complete the original equation, we can rewrite it to find the constant term:
\[
x^2 - 14 + \_\_ = 0
\]
This suggests we solve for the blank that will satisfy the equation \( -14 + \_\_ = 48 \).
Thus, we find:
\[
\_\_ = 48 + 14 = 62
\]
Therefore, the number that completes the equation is:
\[
\boxed{62}
\]
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